Generating a Genuine Mathematical Discussion

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

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

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

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

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

“Umm….no. What happened?”

“He failed his Calc quiz.”

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

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

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

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

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

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

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

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

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

I don’t know what to do.

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.

It’s Not My Responsibility

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

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

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

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

Is it a certain number of exercises or worksheets?

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

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

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

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

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

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