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?

Twitter Math Makes Me Feel Dumb

Tonight has been weird. I left school with my mind fluctuating between anger, disappointment, and curiosity. I was curious about reflections and the thought of reflecting a triangle over a parabola came to mind, wondering if it would have some sort of fun house mirror effect. The more I thought about it on my commute home tonight I realized that derivatives would be involved, so clearly this wasn’t going to be something I could try to do in Geometry.

Then when I checked Twitter tonight I saw this.

<blockquote class=”twitter-tweet” data-lang=”en”><p lang=”en” dir=”ltr”>Reflecting over a circle <a href=”https://t.co/WaqQ04G5ji”>https://t.co/WaqQ04G5ji</a&gt; <a href=”https://t.co/1zzqMf1nKh”>pic.twitter.com/1zzqMf1nKh</a></p>&mdash; Christopher (@Trianglemancsd) <a href=”https://twitter.com/Trianglemancsd/status/837071141873164290″>March 1, 2017</a></blockquote>
//platform.twitter.com/widgets.js

Low and behold, nested within the Twitter stream was a question involving reflecting parabolas. I thought it was neat to watch the discussion unfold in front of me, but when I came across the term normal line, I felt stupid. I had no idea what a normal line is. I don’t know if I never was introduced to the term, or I had forgotten, but I do know it’s not that I was rusty. So rather than try to participate, I just kind of shut down because it made me feel dumb.

The problem I have is that I am the math zombie that became the teacher. I finished a math degree and didn’t really know math. I discovered I was a math zombie when I first began teaching Algebra I and I couldn’t answer a typical mixture problem. (If I had a book with me I probably could find the exact problem since it was that scarring.) Any math that I actually know has been more or less self taught with the aid of textbooks, YouTube, perseverance, an enormous debt of gratitude to the History professors at BGSU who challenged my conception of knowledge, and the students who drug me through. Hopefully that doesn’t mean I am dumb, but that does mean that my mathematical knowledge is extremely piecemeal and lacking in formality. Some of the reasoning I try to share with my high school students clearly lacks the rigor of proper mathematics, as has clearly been pointed out on occasion, but I can confidently say it is mine. Sometimes the responses sting though. I was so excited to share my explanation about rationalization with the world, but was dismissed by some because of canonical forms and the definition of radix. I had to look up canonical forms (which made me want to flip that guy the bird) and I’m still not sure what a radix is or how it impacts square roots. Kind of rips the confidence straight out from under me.

But as painful as times like these are, it helps remind me what it must be like for my students. I can empathize with ALMOST every single student in class to some extent, at least in the attitude towards academics, because I have been there. Math wasn’t, and still isn’t, always easy for me, I need moments like tonight to remind me of that. When I leave work angry and disappointed because of student work, it’s night’s like these that I remember what it was like…

to be worried about grades first and foremost.

to not wanting to focus because other assignments are due.

to just not being able to think about math because, well, just not today.

to struggle to try and remember all the steps in this witchcraft.

to look at a quiz and think, “We didn’t go over that!”

to wanting to just get by and get done.

In a perfect world all my students would come to me with amazing prerequisite knowledge and be highly motivated to learn. That’s not the world we live in though. Without empathy for all the situations our students find themselves in, to many of us wind up browbeating kids into obedient behavior, which just breeds a culture of compliance. My hope is that with some understanding and a little patience I can get a student to want to contemplate the reflection of a triangle over a parabola because…, well,…why not?

I Wanted to Write a Math Post…I Really Did

Here’s a scenario for you MTBoS, can I rotate a point on the coordinate plane 53 degrees without the aid of a protractor or some sort of technology beyond a scientific calculator? I had been wondering since the day I told my classes that they wouldn’t have to worry about knowing anything beyond a multiple of 90 degrees. On my drive home I started to visualize ways that trig functions could do the rotation of any angle and have been wanting to try it out during class.

I found an opportunity after school today to attempt my rotation frustration with a student. She took a different approach then I would have. She established where a 90 degree rotation would have been and then did proportions to figure out the new point. For example, we used (3,4) rotating to (-4,3) and since 53/90 became about 58%, she used 58% of 7 (the distance between 3 and -4), and found the rotated x value to be about -1.1.  As we took time to accurately graph what she was doing, we noticed that what she had created was a right triangle with legs of 7 and 1 and she was finding a specific position along the hypotenuse of the right triangle. This was different than what I had found using trig functions to rotate the original triangle formed by (3,4). In contrast to her, I had formed an arc, not a line. This led us to contemplate what is actually meant by rotation, of which we didn’t draw any firm conclusion. It has also lead me wondering if there is a way to make our answers match, can I extend her hypotenuse point out to meet my arc?

It’s discussions like the one above that I live for as a teacher. I think it is the goal of many math teachers to make their students think like that. I think it is the desire for OCTM, NCTM, and even CCS to get students to that point. The problem is that it is really hard to grade that dialogue. It was a dialogue without a designated answer, in which no firm conclusion was drawn, yet so much learning occurred. But it was so wonderful that I dream of the day that I can get a class where that is the norm for 179 days. I have that dream because I have had that class in the past and I know that is how school can have some sort of lasting impact. Those wonderful dialogues are part of a mechanism that can help change and challenge a student’s emotional intelligence.

A couple of years ago I had fortune to have a math textbook with a misprint in the answer key. We dubbed it the impossible problem from the blizzard bag. It was a problem that required the use of logarithms. I gave it to a couple of students to attempt throughout the day, and two of them perfectly illustrate my understanding of emotional intelligence. One student was in his fourth class with me. He had bore the brunt of the harassment I call teaching in at least two of his four classes. When he was presented with the impossible problem, he solved it correctly, saw the answer in the book, and then explained why the book answer was impossible. The other student also solved the problem correctly, but when she didn’t get the answer the book had, she redid the problem two more times. At that point she gave up, frustrated that she couldn’t figure out what she was doing wrong.

The first student, I don’t think he ever held the stereotype of being a genius (sorry if you read this and figure out who you are, I really do think highly of you), but he had embraced the challenges that I had thrown at him about justifying everything, about making sure stuff made sense, and when the time came to claim his authority, he did. The second student, who probably has a higher IQ, never learned to explain and justify her answers. She never learned to claim authority over knowledge always relying on some external force to reaffirm truth. Students like this are ultimately subservient to the textbook or the teacher. I realize that maybe it’s just math for some of these students, but it is frequent enough it does make me worry.

It’s hard to imagine students who come into class and demand to be complacent and feeble minded. Why would anyone want that? It’s not that students want to lack authority, but they are making an economic decision. They are smart enough to know that their worth is measured by two numbers, their GPA and their ACT (or SAT) scores. They want to go to college, and they know that colleges just plug those numbers into a matrix that will then tie a dollar value to the student. The numbers themselves have more value than the knowledge that those numbers represent.  That is why students will ask me, “Is this going to be on the test?” Or my personal favorite, “Do I need to know this,” implying that much of what I teach is actually worthless.

The student with whom I was working on rotations was not there for that purpose. She had come into work on ACT math, of which three problems stand out.

  1. Sequences – She caught on quick, it was the notation holding her up. That is entirely my fault for not showing the notation in her previous classes.
  2. Matrix Multiplication – I haven’t done matrix multiplication since college and wasn’t introduced to it until my junior year in college, in a Linear Algebra class. Why is this on the ACT?
  3. Graphing on the Complex Plane – I have never done this, ever, at any level.

All three of those topics are very specific and can be memorized with very little understanding. Memorizing enough of those has thousands of dollars worth of value. What she did with the rotations, while fascinating and enlightening for me to watch how her mind works, has no immediate impact. Why do we seem shocked by students who like plug and chug math?

So MTBoS, we preach mathematical thinking, growth mindsets, grit, and any number of ideological approaches that hopefully will create enlightened problem solvers, but our students live in a world where they are valued upon correct answers, not original thought. Math as I know it, is essentially useless to many of my students, but the right answers have thousands of dollars worth of value. How do we show them empathy for their plight, but get them to embrace our ideals? I ask you MTBoS because I am losing my students.

Am I Jaded or Cynical?

I am tired. So very tired. I have spent the last seven hours working on stuff for my upcoming evaluation and am yet not done, but I need a break. Part of the reason that I am not done is because I find myself losing the passion for what I am doing. For a break I am going to do a piece of self reflection. I want to decide if I am jaded or am I cynical.

As long as I can remember I have always leaned more towards a pragmatic, pessimistic view of the world. But the past three years (2015-present) have been really weighing on me. I think I have become jaded by my work. Jaded is defined as being fatigued by overwork or being made dull, apathetic, or cynical by having seen too much of something. I’ll ignore the idea of cynical for now, but I would say that I could be considered fatigued or apathetic from having seen too much.

Too much of what though? Too much of everything. Too many rules. Too many rubrics. Too many evaluations. Too many detentions. It’s to the point where all I am starting to see at school is a place not of education, but a place of complacency and obedience. Which is starting to make me feel rather cynical.

A cynic is someone who is distrustful of sincerity and integrity of human intentions. If I am truly a cynic, I don’t believe people are motivated by altruistic intentions. A quote attributable to the great George Carlin, “Inside every cynic is a disappointed idealist.” I think that perfectly sums up my life for awhile now.

The class of 2013 left me with regret, I felt like I didn’t do enough. The class of 2014 left and I felt uplifted. I took some chances, pushed some students to the breaking point, and tried to embrace education for the life altering experience it can and should be. School shouldn’t be a place where subjects are learned to simply barf back answers on a test to get a decent scholarship. That’s why I felt regret after the class of 2013. I had most of those students for two or three years, a couple for four. I should have found a way to push them beyond choosing the safe path. Sure, they weren’t likely to fail, but some of them could have done more and chose not to out of fear. In 2014 I pushed convention, both with my pedagogy, rules, and role. I think I finally had the lasting impact that is supposed to happen in education. Some of those students said the kindest things to me after they left. One even took the time to write a very thoughtful thank you note that I routinely reread anytime a bad day has made me question my purpose.

I started 2015 actually excited for the school year, in spite of losing my beloved Algebra I class. I eagerly shared all of my insights from the previous year. As I began to share them I was met with the opinion of from the local educational service center. I was asked what parents would think. I was told that students are lying to me. My assignments and methods were questioned.

At first I chose to ignore them, thinking what would administrators who never have interacted with my students know. But then I was shaken. It was during a Pre-Calculus class.

At my school Pre-Calculus was quasi elective. Students had to take a fourth math course, but they didn’t have to take Pre-Calculus. I had experienced great success the previous year in my elective math classes by abandoning grades. I told the students that as long as they learned I would take care of the grades. I placed a large amount of faith in my students because to abandon grades meant I didn’t have the traditional documentation found in a normal classroom. I still remember the day it hit me like a rock. I asked a general question to the class. A typical cell phone addicted, vocal student started saying numbers. I would respond with a no and the students would blurt out another number while looking at the cell phone. Finally a friend spoke up.

 

“Stop blurting out stupid answers.”

“Well, I don’t see you trying.” (Still scrolling on the phone.)

“I’m trying to think of a decent…”

I decided I should intervene at this point, “I’m not just teaching math, I am trying to teach you behavior as well. People can’t multitask. It’s impossible. You are proving it right now. I counted at least 43 times you glanced at…”

“Yeah, whatever, you said that we would all probably get a B or higher anyway.”

 

I froze in absolute frustration and disgust. It’s what I do as a public school teacher, when I am silent it means that all I have going through my head is a string of cuss words and other obscenities to call my students.

That was the beginning, but it wasn’t the last time. I got tired of being lectured about bell to bell instruction, learning environments, resources, and the gambit of teacher speak. As I kept hearing it from the experts, I started to notice it from my students. They seemed to be tuning me out more and more. They seemed to be taking advantage of me more and more. I started to question their motives. I started to think that they don’t care about learning, all they really want is the plush transcript, or the good GPA, or honors sticker on the diploma.

I’ll go back on Monday and reread that note and wonder if that year was just an aberration. Maybe it was just a perfect storm for me to succeed, just the perfect mix of the right students, with the right administration, and right environment, that fit my personality and beliefs.

Right now my beliefs and convictions don’t seem to mesh with what my environment wants from me. So I’m tired, which I guess means that I am jaded. As I have become more and more jaded I have noticed myself becoming more cynical. I start to see nothing but obedience and compliance around me, even though they haven’t changed, I have. It makes me worry about my job security. I guess that makes me cynical.

Grades and Empathy

My students are just finishing their last rounds of state-mandated testing. Many of them are burnt and fried. It’s just too much testing all at once, especially for the sophomores at my school. For my tests the results are mixed.

I fall under the auspices of something called Student Learning Objectives (SLO) under the Ohio Teacher Evaluation System (OTES). I have to create a test that is to represent a years worth of material and administer a pre-test and post-test to show growth. Since the test is practically identical, much of the material is new to the students. To encourage students to take the tests seriously, we are allowed in our district to use the SLO as an exam grade.

Without the threat looming of exam grades, the only consequence SLOs had was teacher evaluations. To put it another way, the test wasn’t necessarily a measure of student ability, but of teacher quality, or in the case of most SLOs, teacher test writing ability.

Most of my SLOs are completed and the grades are mixed. Overall though, I feel that they are too low to use as an exam score that would accurately reflect what was accomplished during the year. So why are the scores so low? I think there are three factors in play.

First, students just flat out forget a large quantity of the information they are presented with throughout the year. I have read psychology research, cognitive load theory, and numerous other theories as to why this occurs. I believe there appears to be debate about whether instructional practices or student attitudes account for this phenomenon, but either way it exists. Our students just forget so much stuff.

Second, so much of math instruction is perceived to be this, whether it actually is or not.

What happens is that students freak out and have borderline panic attacks when problems don’t match the memorized examples from class. Students who normally volunteer information and come up with some of the best ideas in class shut down when the assessed problems don’t match their memorized examples.

Third, my usual assessment format does not require to students to be as attentive to precision as they should be. The multiple choice format SLO I gave requires precision regarding negative signs and arithmetic. My usual assessment rewards creativity at the expense of precision.

So why are the scores so poor? The first problem, students just forgetting, I think that is something that is only minimally impacted by teachers. I can encourage, I can try and provoke, but if students won’t authentically engage with the material learning will not last. The second problem I think I usually do rather well with, or at least with the grade obsessed students. My reading through pseudoteaching has really changed the approach I use to the presentation of my lessons. I have adopted a less is more approach in my lesson presentations, emphasizing that the work done in class only illustrates concepts and that can be applied in many different scenarios. And I am pretty happy with aspect of my assessments as my students have become more flexible and adaptable in new mathematical situations.

But the third reason why I think the grades are low, the precision, is something I need to change. In the multiple choice section of the SLO, I noticed that many students had the concept down, they were just making procedural errors. That means, to some extent that the low scores are my fault. Every year I keep saying I will, but because my open ended assessment reward creativity more than precision, it is ultimately empty rhetoric. I don’t want to just dump my current assessment as I am happy with the outcomes. I was able to use the students attachment to grades to make them be more mathematically creative. I think I can do the same by using some multiple answer assessments throughout the year. It would force the students to become more accustomed to mathematical accuracy and lingo.

Now back to the original purpose of this post, curving grades. I have never felt the urge to curve grades like I have this year. I also never realized there were different ways to curve grades. In past years I have felt that the exam scores accurately represented my students knowledge of math. As I have interacted with these students over the course of one to three years I have a pretty accurate representation of their mathematical potential regardless of their specific exam score. This year I had a couple of students perform much more poorly than I expected and much of that performance is based upon my not training them well enough to handle the precision of a multiple choice exam, hence my urge to curve the grades. So here is a list of all the questions and dilemmas running through my head.

  • While I have sympathy for those students who engaged fully throughout the year and want to take blame for their poor performance, I have a handful of students that have so effectively tuned me out that I really don’t feel the need to curve their grades. In a way I want those students to suffer the consequence of lacking authentic engagement, which in this case would be a drop of a letter grade or possibly two. For clarification, none of them would be in danger of failing, just GPA reduction.
  • I thought about applying the curve to only those students who have shown effort throughout the year, but I revolt at that for two reasons. One, I despise effort grades. Two, if I pick and choose which grades to inflate it ultimately renders the concept of an exam mute.
  • I realize that much of my desire or lack of desire to curve is based upon which math class students are in. I have more sympathy in my required courses (Algebra 2) and less sympathy in our elective courses (Pre-Calc).
  • I have a couple of curve breakers whose scores are high enough that it renders the curve pointless for my low scoring students.
  • I really don’t want to put my students through another exam. I don’t want to take the time to make another exam. It rewards those students who didn’t take the SLO seriously enough the first time around knowing that there is potential for another exam.
  • If I give another exam it punishes the students who did well on the SLO.
  • If I let the students just keep the higher of the two scores, why stop at just two? Why not give the students three, four, or even more opportunities? And if I give them endless opportunities isn’t it really just like me determining their grades subjectively?
  • At the end of the day my students did well enough to help ensure my job security (met SLO growth targets). Well, most of them. I think there were a couple that really want to get me fired. Is it wrong that I want to somehow manipulate the grades in a way that either rewards, or at least doesn’t harm my students grades?
  • I really, really want their input. However, I want their input in a manner that takes into account more than their individual grade, which I feel most  are capable of doing. I do fear the implicit pressure placed upon me to “control” my class and that requesting feedback from students is empowering them too much.

Isn’t it nice that grading is so simple.

Why I don’t do homework

I hate homework. For the few people that know me, I feel like I have at least an average vocabulary, I would pretend that it is above average. And I cannot stress the importance of the choice of words. I hate homework. I am not annoyed with homework, I am not repulsed by homework, I am not disgusted with homework. I hate homework. There are two major misconceptions about homework that I would like to attack to explain why I hate homework.

But first a clarification, homework can have a purpose. Readings that provide students with information that will be necessary for participation in a class discussion. Using flashcards to memorize medical terminology prefixes and suffixes, those are good uses of homework. So why do I have a problem with homework? Here’s why.

Homework reinforces the concepts covered in class.

If I ignore whether an assignment is graded based upon completion or accuracy, I see eight possible outcomes from the homework assignment. If a homework assignment is to achieve its maximum benefit it needs to be in a place where a student has mastered the material enough to complete the task, but still has to expend intellectual effort. The goal for this homework is that the student is moving recently learned information to become part of a long-term schema. Once in schema, the material becomes intertwined with prior knowledge and can be used to interpret and understand even more concepts.

Publication1

However, very few homework assignments covered in class actually fall into this sweet spot. If all outcomes on my flowchart have equal probability (which is a big assumption) only one out of eight homework assignments will have the desired result. Only one out of eight outcomes will reward intellectual effort positively. Seven of the eight will reward aspects of school that do not help us learn.  Imagine homework for a student, with the student thinking, “that was dumb, that was dumb, that was dumb, that was dumb, that was dumb, that was cool, that was dumb, that dumb.” Now repeat that over and over and over for twelve years. Is it surprising that so many of our high school students have a negative view of the purpose of school?

Students need something to work on at home.

Often I will get a question from a parent the first time we meet. Usually it goes something like, “What should my child be working on at home?” Deep down I really want to tell the parent that their children should be working on whatever interests them. In reality though, I get a sense that work is associated with learning. If a large math assignment is completed, much learning has occurred. Big assignment equals big smarts (insert caveman imagery here).

But work is just that and nothing more, work. A better option for parents who are concerned with their children would be to simply have them demonstrate what they are learning in class. What follows might come in fits and bursts, with some information being comprehensible and mastered, while some of it might lack clarity, but that is the nature of learning. A student working doesn’t necessarily mean that the student is learning. Learning takes work, but not all work is learning.

So what’s the solution?

I wish there were an easy answer, but there isn’t. Research is mixed, even personal experience is mixed. So what I have chosen to do is have my students do as much work in front of me as possible. At least that way I know what they can and cannot do. I will list a practice assignment for students to work on if they desire, but it is optional. It is optional precisely because the student would need to see the benefit of homework, how homework’s purpose shouldn’t be grades, but rather homework should exist to benefit understanding and mastery. Not many take advantage of this opportunity, but a few have.

I hope more will.