A (brief) farewell to the Memory Enhnacer

RIP Memory Enhancer - February 14, 2015

I laid to rest recently an assessment I have been using in some form or another throughout my teaching career. Memory Enhancers I called them because a desired outcome was that students enhanced their memory of mathematical ideas of days past. They were known as Review Sheets once upon a time and were derivations of the Friday Sheets that were used by all teachers at my first school. The department chair at that school developed Friday Sheets as part of his masters degree and decreed that all teachers under his purview would use them.

I liked the Friday Sheets primarily as an alternative to tests. Some teachers base grades solely on tests and quizzes. But what are quizzes other than mini-tests? Friday Sheets were take-home assignments that reviewed previously learned material and students could work on them over a week. Handed out on Friday and due the following Friday. My department chair had designed them to be sequential in his courses, so they built off one another and were another way for students to improve their understanding.

I did not like having one every week. It was a lot of grading and it was on top of the work the students were already doing. Students usually waited until Thursday nights to do them and cheating happened. When I changed schools, I renamed them Review Sheets and made them less frequent. I started using Review Sheets also as a place to ask longer problems, problems that were not really fair in a timed test setting. They were also a place for me to experiment with different types of problems and ones where students could work on graphing skills.

When I changed schools next, Review Sheets were reborn as Memory Enhancers. I started to use them as another way to present new ideas. Students were asked to deal with ideas that had not been discussed in class. I wrote open-ended problems where answers would not all be the same.

I carried the Memory Enhancer with me to my current school and started focusing anew on them. I tried to make them sequential so that students would learn one idea in Memory Enhancer #1 and that idea would be further developed in subsequent Enhancers. I even started having students do them in pairs. I was able to write some problems I really liked and students had usually 10 days or so to complete them. They were encouraged to see me for help and I highlighted time when I was available.

Some students have loved these assignments. Some have recognized that here were assessments where test anxiety need not be a factor. They could work on them over multiple nights. They could spend 20 minutes on a problem, figure out that they had traveled down the wrong path and then go back and start over and not worry about running out of time. They could use their books, their notes, and old homework.

But, here and there, students would succumb to temptation and cheat. The inevitable awkward and uncomfortable confrontation between teacher and student would ensue and often lying was used a means to cover up plagiarism. Due to the nature of the assignment, administrators were unwilling to respond too severely; they believed we could not prove the cheating if students did not admit to it. Me? We are not a court of law. We are a school. We are here to educate. Part of that education in my mind is teaching a student to be honorable. I tried different tactics, including having them write an honor pledge (based on the Davidson College Honor Code).

This year was rough. With each Memory Enhancer, I found myself doubting the originality of the work of multiple students. By the time I got to #5, I was questioning more than 10% of the papers turned in. This is after having confronted multiple students. Some of the questioned papers were repeat offenders.

The frustration this was causing me was too much. Enough, I said, and I made the decision to not assign them again this semester. I was sad over this decision because I knew there were students who saw them as a saving grace in contrast to the big bad tests. I do think my students are missing out on their education by not having these to do. Most students probably don't appreciate this. Most are probably happy about this decision.

I am planning on re-introducing them in the fall. This was by far the worst of the copying I have encountered, so I am expecting it is an extreme example and not a sign of things to come. I will do things somewhat differently and maybe that will help. I want to help students learn about honor and trust and ethics and I think these assignments can be a means of doing that. But I won't wait until February next time if they start frustrating me. It's just not worth my health.

The Portfolio - Incarnation 2014

In art courses, students often develop a portfolio over the course of a semester. Students place all of their work in the portfolio, or maybe just the best of their work. The idea is to have a collection of art that, when considered together, is meant to represent a student's output for the semester.

I really like this idea as another means of evaluating a student. I like that the students choose what work goes into the portfolio and that they can make decisions such as layout and formatting and the like. I think it might encourage students to be proud of their accomplishments. Also, by it being a collection of work throughout a term, students get a chance to show off their development.

In my vision of the typical American high school math class, students don't get to choose what gets evaluated by their teacher, nor do they have the opportunity to design their own assessments. Rather, all students take the same assessments (tests and quizzes largely) that were created by their teacher (or by some publishing company) and a student is judged by their work on these. A student's semester grade is often determined based on how she or he does on these assessments.

I am establishing a new course this year at my school: Math 2 Standard. This is an integrated course that is much more similar to our honors math courses. I believe the integrated aspect alone is reason enough for the change as I believe (and I think research supports this notion) that an integrated math course produces better learning outcomes than the traditional American math progression of Pre-Algebra, Algebra 1, Geometry, Algebra 2, Trigonometry, Pre-Calculus.

But, by having a new course to design, I made the decisions to do some experimentation, so I dug up the Portfolio. I had attempted to have my math students create a portfolio in years past, most recently at my last school. But, that was just them collecting all of their work and storing it in a three-ring binder that was kept in my classroom. I never ended up assigning a grade to that portfolio, nor did I really have the students do anything with it.

In the fall semester of this year, however, I gave much more detailed directions. Students were to find a certain number of problems (not entire assignments) that they had already submitted for evaluation, and were to revise their solutions in an attempt to improve. I have long championed mistakes as being useful, that mistakes are necessary in the learning process. I suggested that by reflecting on mistakes, students can improve their understanding and thus establish true learning. So, the Portfolio was an attempt for me to put my money where my mouth had long been. I would evaluated a collection of work, work chosen by the students, that they offered to me as evidence of their improvement. As a part of the Portfolio, students were to write a page or so about their process: how did they improve on their work?

For the most part, I feel this project was very useful and productive. Looking over some Portfolios, it looked like some students truly improved a lot, that they took the project seriously. Some students, however, appeared to have just copied the work of others and taken the path of least effort. But, all in all, I was glad I did it and I do think it was a valuable learning experience. It was definitely better than a final exam.

Unfortunately, it produced an immense amount of work for me to grade. It was as if 58 students took 58 different final exams and wrote 58 reflection essays, all of which I evaluated during my holiday, ahem, break. By far, this assignment took me longer than any other to correct. So, I won't be doing it again in exactly the same format. Instead, I will evolve it to make it more practical for me.

This semester, there will again be a Portfolio. This time, I will limit it to 2 or 3 large-ish problems and students will write a reflection on each problem and how they went about revising it and how they improved their learning from it. I really like the idea of students choosing for themselves work that they want me to evaluate. I really like the idea of them going back and re-doing problems, trying to better understand the ideas and the processes and what it all means. So, if I can keep up with my blog, I will write again about the Portfolio in 5 months' time. I wonder if I will do it again next school year...

Striving for the "acceptable" grade (Part 2)

When I last left you, Jolene had been wanting to switch math courses because she wanted to be able to get an A in math and spend less time achieving that grade. Our plan was for Jolene to take the next test and see how it went and she would decide based on that result.

Jolene earned an A- on the Unit 2 Test. She decided to stay in the honors course.

I know, this is kind of a quick ending to a long story, but it was really as easy as that. Why? Because there was only one important thing - her grade. There was no discussion as to how long she spent preparing for the test.

A few weeks later, she remains in the class and her grade is at an A-. But there's been no discussion involving me about whether her other classes are suffering or if she has, in fact, reduced the amount of time she devotes to her math homework.

Today, we had our Unit 3 Test. I have no idea how Jolene did and I probably won't look at the tests until Thanksgiving break. (Yes, I do school work during my "break". In fact, most teachers I know use their breaks to do work. Often a significant amount of work.) At the end of today's test, Jolene commented that the test was difficult. But, she didn't leave my room crying.

No, today it was Brenda who left crying. Brenda, my superstar. Brenda, like Jolene, is a new student at my school. She considered trying to move on to the third-year honors course. In the end, she opted to stay in my second-year honors course, and this has proven to be a pretty good fit for her. Her grades have often been near or at the very top of my class.  But, she has been learning a lot and the class has not been as simple as she had thought it might.

Today, Brenda melted down at the end of the test. Time was up and just as Brenda was about to hand her test to me, she noticed that there was a page 6. She looked at me and asked for time to complete page 6. I told her no, that I had not been able to give students in the other period extra time, so it would not be fair to give her additional time. Moreover, the last page had one problem that was worth 2 of a possible 34 marks. There is absolutely no way that 2 marks will have any impact on her semester grade. Brenda has already demonstrated to me remarkable mathematical skills. She readily recognizes pattern and is proficient at explaining how the ideas connect.  Despite all of this, Brenda broke down crying over not having gotten to do anything on page 6.

I'm not sure about this, but I'm guessing Brenda's breakdown has many more reasons behind it other than the math test she just took. Regardless, not finishing the test ignited her feelings and brought them to the surface. Since I don't know what's going on with her beyond my math class, it's not fair to say that the almighty letter grade is the main culprit here. But, the letter grade certainly has played a major role. It's as simple as this. If the test had no grade associated with it, would Brenda have left my class sobbing today?

Keeping up with the Joneses

In math, we say that 45 > 42 because 45 represents a larger quantity than 42. So, it seems fair to say that if Peter gets 42 points on a test and Claire gets 45 points on the same test then Claire did better than Peter. Right?

Hmmm... Suppose there were 50 points possible on my test. Suppose there are 10 questions and each is worth 5 points. Then perhaps Claire got 9 problems completely correct and 1 problem completely incorrect, and Peter got 1 point off of 8 problems. So, who did better? Is one large error better than several smaller errors? What seemed so clear now becomes murky.

In the "safe" world of adding up points earned and dividing by the points possible, Claire gets an A- (90%) and Peter gets a B (84%). But, if the above were true, it seems challenging to me to try to decide who did "better".

Keeping with the above scenario, Peter would have two "perfect" problems and 8 "imperfect" problems. What if Peter got full points on the problem on which Claire earned zero points? So, he understands that idea way better than Claire. That would also mean that Claire understands 8 questions better than Peter, but only in a small way on each.

Suppose, instead, the test were out of 214 points? How much of a difference is 3 points?

Comparing students to students is the real issue here. We, as teachers, do this all the time. The SAT does it. Colleges do it. Students do it. Why? To see who is better, or worse, or the same as us? How does that help my learning?

Let us keep Peter with his 84% and Claire with her 90%. What do these scores tell us about how they are doing overall? Not much. Those scores provide us only some measurement of how they did on one individual test. Perhaps Peter has never scored as high before. Perhaps Claire has never scored as low before. Now what? Does that diminish the value of Claire's score? Does that augment Peter's? Might we now say that Peter has, in a way, done "better" than Claire?

No, I go for Option Q2. With such scores on a test, do not be concerned with such questions as "Who did better on this test: Peter or Claire?" Rather, ask the question "What does Peter's performance on this test tell us about how Peter is doing at learning the ideas being tested?" Ask the same question of Claire. But don't bother with trying to make sense of comparing the two.

Using rubrics - The letter grade on steroids

In my last post, I began discussing alternatives to the standard letter grade and mostly wrote about using narratives for evaluating work. Tonight, I'd like to discuss the use of criteria and grading rubrics.

Criterion-based grading, for me, is a large improvement on the standard math teacher grading system (i.e. add up points earned and divide by points possible to get a percentage that converts to a letter). I use criterion-based grading exclusively in my Senior Stats course. Here is an example of grading rubric I used this year.

Here is how I use my rubric. I read through the project as a whole, making notes and comments as I go. I then evaluate each criterion, one by one. Each criterion has various achievement levels. Some go from 0 to 2, others from 0 to 4. In order to reach a level 2, a project must satisfy all of levels 1 and 2. To reach level 3, a project must satisfy all the requirements of all 3 descriptions.

I attempt to write the descriptions as plainly as possible and in a logical progression so that attaining a higher level should show an improved understanding. For example, take my "Displays" criterion. To reach level 1, a student must only create one correct display that is relevant to the project topic. Level 2 is attained if a student makes multiple types of displays. So making 2 different histograms would be insufficient. Making a histogram and a boxplot would satisfy the description. In each case, the displays must be made "correctly," meaning, in part, that they should be accurate and be labeled sufficiently. To reach level 3, the displays must "communicate well" and have a "high level of accuracy." Essentially, this is a way for me to distinguish between the student that makes a few sloppy displays and the student who makes a few excellent displays. Finally, level 4 requires a certain degree of sophistication within the displays. This means that they are all effective and lead to insightful analysis. There is a level of complexity within the displays.

So, I evaluate the project against all of the criteria. I write narrative comments for each criterion to explain why a certain level was awarded. I note errors as well as strengths. This example will give you a taste of what I try to do each time (you can zoom in on the image to make it large enough to read).

In the end, however, by using achievement levels, what I've really done is assign a variety of grades to a project. In a sense, I've given a student a "letter" grade for their introduction, another for their data, one for their calculations, and so on. So, I feel like I'm not exactly moving away from the letter grade entirely. And, given where I teach, in the end, I do have to assign an actual letter grade to the project. So, I do my voodoo and do that. More on my voodoo in another post. I'm tired. Good night.

What the heck is this blog all about anyway?

A major focus of my professional development in the last ten years or so has been the role of evaluation in the education of high school students. Hi. I'm a high school math teacher. Want more? I am a math teacher at a prep school. At an intense college preparatory secondary school. So, kids are sent here to go to college. Or so it would seem. And I am part of that.

Now, I want my part at my school to not be just as a tool that students use to get to the next place. No teacher wants that. So, I question everything I do, and the biggest question that looms over me, especially at the end of each quarter is what the heck am I doing assigning a grade to these kids. I mean, how does giving a student a grade really benefit that student?

I wholeheartedly (yes, I mean that, with every ounce of my heart) reject the "positive reinforcement" idea. Quickly, if you think negative reinforcement is bad, then why should positive reinforcement be any better? Both are used as a means of obtaining compliance. Don't do this, or you will go to your room. Do well on this test and I will give you a token that helps you get into college. Same thing. Don't trust me, trust Alfie Kohn who exhaustively researches his writing. Check out this article first.

But, I am obligated to pursue this practice given where I currently teach. Really, this would be the case at the vast majority of secondary schools, so it's hardly surprising.

At the end of each quarter, at the end of each project, at the end of each test, and so on, I assign a letter grade to evaluate the quality of the student's work. This simple letter grade only conveys how one has done compared to one's peers. It does not offer any suggestions as to what is "good" or what "needs improvement" or if a student has improved or regressed or been stagnant. And, given the high stakes involved - Yale? Carleton? UC Santa Barbara? - students learn that these tokens are what they need so that is all they pay attention to. Yes, I am generalizing, but it is a valid generalization.

My proposal: Eliminate with great haste and with great contempt the letter grade. Evolve as a teacher, evolve as a school, and evolve as a student. Who knows, maybe evolve as a society?