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?

Striving for the "acceptable" grade (Part 1)

I had an experience recently in my honors math class that reminded me of this blog. So here I am again. But only for 15 minutes. Go.

I got an email a few weeks ago from our registrar saying that Jolene (not her real name, of course) was dropping the honors course and changing to the Algebra 2 course. "Wow," I thought. What was this all about? My gut feeling was that Jolene was doing just fine. Maybe not at the very top of the class, but certainly in the top half. (And I suppose it's contradictory of me to have used my grades as a way to think this through. Do I contradict myself? Okay, then I contradict myself.) From my observations of Jolene in class, I certainly did not notice her struggling to the point of giving up. So, to check on my gut feeling, I checked my online grading program (my school requires me to use one to my revulsion) and discovered that her grade at the time was an A-.

I immediately emailed Jolene's advisor writing that this was not a good idea, that Jolene was doing very well from a grade standpoint and according to my observations and impression of her in class. (As I only had 12 students in that particular section at the time, I very much trust my classroom impression.) Jolene's advisor, Liz (not her real name), wrote back saying that Jolene was spending so much time on her math homework that she wanted to drop the honors course so she could devote more time to her other classes. Jolene had been relying on other students, it turns out, for a lot of help, and always worked to get a perfect homework assignment.

This, on the surface, sounded like a very good reason to change courses, but, I replied back and suggested that she could likely spend less time and still earn a B. Furthermore, homework assignments in my class are not graded and I have no expectation that a student get every question correct by the day it is due. Rather, my expectation is that the student do as much as they can on each problem and get help as needed. (I had other discussions with honors students around the same time that made me realize I had not adequately communicated these expectations to my student.)

Of course, the main sticking point was the grade. I found out that Jolene's preference was to switch courses so that they could get an A in math and have more time to spend on other subjects. She mainly wanted to switch because she was concerned she might not be able to get an A in the honors course.

Jolene, Liz, and I met to talk this over. I emphasized to her that she seemed to have a very good grasp of the ideas, that her results were indicative of this, but that it was certainly possible her grade could be a B. I did my best to emphasize that I did not think her getting an A was paramount. My opinion was that the Algebra 2 course would be too easy and she would be bored and that, in the end, this would lead her to a less rich education. Liz, her advisor, supported me on all of this, and we devised a plan: she would take the next test before switching and use that to make her ultimate decision. (Again, something that was grade-dependent, but at least something the student would identify as tangible.)

What happened? You'll have to wait for my next post as my 15 minutes are up.

I am the Gatekeeper.... are you the Keymaster?

The role of the letter grade as gatekeeper is yet another reason I am looking for something better.

At the end of my honors 9th grade math class today, one of my students asked me about moving up next year to the accelerated 10th grade math course (yes, we have a level above "honors"). This student, let's call him Ricardo, is a new 9th grader, that is, he was not in my school's 8th grade last year (my school is an independent K-12 school). He spoke of how when the year began, he felt mostly lost because the math curriculum at his last school was far behind ours. So, he explained, though he began the year in the C+/B- realm, his grades have steadily improved and so has his confidence and understanding of the material.

I agree with Ricardo's assessment 100%. However, his semester grade was a B and he earned a B on the semester exam. Our criteria for a student moving from the honors course to the accelerated course is a 95% and a student ought to have earned that without extraordinary effort. Essentially, for the student who wants to move up, the honors course should be easy.

At this point, it does seem like Ricardo is understanding the newer topics with more ease than was the case in the first quarter. In fact, his level of performance at the moment might be at the necessary level for the move he wants to make. But, is that enough? From a pedagogical point of view, if he shows a mastery of the course throughout the second semester, should that make up for his difficulties in the first semester? As the topics in the course build off of each other, should he show such a mastery in the second semester, it would show that he has now mastered the first semester topics, too.

However, from the grade's perspective, it may be too little too late (to use a boring cliche). His semester "average" was an 85. Thus, even if his average in the second semester is 100, he would average 92.5 for the year. He would fall short of the 95 benchmark. And, from his own admission, the first semester was not easy for him.

How much importance should really be put on that grade?

I do think our school is flexible enough to look beyond the letter grade and take the student's full performance into consideration. Also, the student could potentially make the move from honors to accelerated between the 10th and 11th grade year.

That's only one small example of how letter grades serve as keys to opening doors in education. The "best" example of this role of the letter grade is college admissions. Ricardo probably sees getting an A in honors math as not enough. Instead, he wants to get an A in accelerated math so that he can say he took the most rigorous courses at our school to impress colleges.

Do you see what I see? If this is Ricardo's thinking (or his parents' thinking), Ricardo is treating his math courses as steps on the way to college. The content of the courses is of no relevance. All that matters is that he does well in the hardest classes. Does he enjoy math? Does it spark his imagination? Does he explore topics in math in greater depth in ways that are not graded? Maybe, but the system does nothing... NOTHING... to promote this love of learning.

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.

If I must give a grade...

As promised, here is a recipe I have followed in previous years for figuring grades on math tests.

I begin by correcting the tests against my answer key (also known as a mark scheme). Each problem has a defined number of points (or marks) and each point is earned for either an accurate value or a a correct method. Many problems have multiple steps, some necessitate multiple methods, thus a single problem can have multiple method points and multiple accuracy points. I use the idea of follow through. This means, if a student errs in part (a), but then properly uses the incorrect answer of (a) to solve part (b), then they are not penalized in part (b) even though their solution may be incorrect.

After I correct all of the tests, I order them with the greatest number of points on top. I then go through the tests and try to determine the quality of the work produced. I begin by setting the grade boundaries. So, let's say a test has 35 possible points and the top score was a 33. I might start looking at tests with 29 points and see if I think they meet the "A" criteria. (My school has a one-sentence description of "A" work. I have expanded greatly on this to explain what I expect students to show in my math class.) Having years of experience, I have my criteria pretty well set in my head, so it isn't too hard for me to think to myself, "This 29 feels like an A-." If there are multiple 29s, then I read over one or two others to see if they also feel like an A-. Then, I look at the 28s. Do they feel like an A-, too, or are they more a B+. Then, I do that with the 27s. At some point, the quality of the work clearly no longer meets the "A" criteria. So, I might decide that 28 is the bottom of the A's. Then, I do the same thing to determine the B/C boundary.

Once I have found the A/B boundary and the B/C boundary, I calculate a best fit line to convert the number of points on the test into the 100-90-80-70-60 scale. Why? Because I have to give grades in that system. That's why. So, I then look at what that does. I look to see that it accurately represents the quality of the students' work. If necessary, I look to see what that does to the C/D boundary. Does that boundary seem reasonable? If anything seems out of whack, I tweak my boundaries. By "out of whack", I mean, are grades being assigned fairly? For example, a 15 might work out as a C-, so I ensure that the quality of work for a 15 is truly a C-. If I think, no, it's a D+, or, no, it's a C, then I adjust my best fit line.

This may seem like a laborious process, and it certainly is in comparison to the standard math teacher system (add up points earned and divide by points possible to get a percentage that converts to a letter), but I used it for several years and became quite adept at it.

I like this system better because grade boundaries are not pre-determined. With the other system, I found that sometimes a test was easy, and mediocre work was awarded a B while in other cases excellent work was awarded a C+. With my system, grades are awarded based on the quality of the work actually produced. With the other system, I have to be a fortune teller and predict what excellent work will look like.

If this system sounds at all familiar to you, then you probably have taught at an IB school. The International Baccalaureate (IB) uses a system much like this to assign grades to their exams. I know, because I used to work at an IB school and the math department graded all of its tests in this manner (minus the converting of the score into the 100-90-80-70-60 scale).

It's not a perfect method. Not all 27s on a test are the same, for example. And, it is quite different from other math teachers at the school where I teach. So, many students are confused by it and some don't bother trying to understand it. But, I truly believe it is in the interest of the student to do it this way.

This year, I am teaching a 9th grade course and, obeying a request from higher ups, I have simplified my process. I often found that when I did the above conversion, a zero score on my tests would convert to a 50. This makes sense to me because in the 100-90-80-70-60 scale, an F technically has a range of 60 points. Why? Who knows. To me, a range of 10 for each grade makes sense. So, making a zero convert to a 50 makes sense.

So, this year, all of my students begin with a 50 and then the points they get on a test are used to figure what fraction they get of the remaining 50. For example, let's say a test has 35 points and a student gets a 25. 25 out of 35 is 71.4%. So that is the percent of 50 they get in addition to the 50 they started with. 71.4% of 50 is about 36. So, a 25 would convert to an 86 (36+50=86).

Okay, I know you're likely confused, so comment away and I will try to explain it better...

The 90-80-70-60 scale... what gives?

In the US of A, it is well known that, on most tests, if one scores between an 80 and 90 percent, one earns a letter grade of a B. Ummm... okay. Why? Why not 60 to 80%? Why not some other arbitrary range?

Here is more of the "standard" scale:

• 100-90% = A - which typically means "excellent"

• 90-80% = B - which typically means "good"

• 80-70% = C - which typically means "satisfactory"

• 70-60% = D - which typically means "poor"

The last of these letters is F which typically means "Fail". Of course, that means that the F scale goes from 0% to 60%. So, you're trying to tell me that the F range is 6 times larger than each of the other ranges? Why is that?

This also means that the range for "excellent" is the same size as the range for "good". Does that make sense?

My point? These ranges seem totally arbitrary and I can see no benefit to the student or to my teaching in having to use such ranges.

This grading scale also begs the question: Percent of what? Now, in math classes, it usually means the percent of points a student has earned against the possible number of points. So, if a test is worth 40 points and I get 30 of those points, then I get a 75% and thus a C. So you're telling me that I understand 75% of what I was tested on and this is satisfactory? What if I that 75% represents the top score on the test? Then what does that mean about my "satisfactory" grade?

Even weirder is that this scale gets used in other disciplines. I write an essay in English. My essay is "good", but not "excellent", so I should get a B, I guess. But, how in the world could one possible assign a number to my essay.

Anyway, I give this post a 77%.

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?