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.

To grade or not to grade

Is it fair to not tell students they are being graded?

Today, my students worked on an in-class group activity. It was an ungraded activity. They were shown a video of a water tank draining out of a spout and asked to figure out how long it would take the tank to empty. I withheld nearly all of the necessary facts, figures, and formulas. They were required to ask me questions. If I knew the answer to their question, I would answer it. For some of the answers, I gave them cards I printed ahead of time with information like the height of the tank or the formula for volume. Some of the information was in the form of a video of the spout filling a 2-cup measure. Ultimately, the calculations involved were rather ordinary and simple: multiplication and division. The activity was designed to help them develop their problem solving strategies and have them consider what information is important and necessary to solve the problem.

Some groups predictably got straight to work and proceeded quickly through the problem and found an answer in relatively short order. Other groups, just as predictably, took their sweet time and spent many a minute idly chatting and not worrying much about solving the problem. Experience told me that some groups just need a bit longer to warm up and realize that, in fact, I am not going to give them the information they need unless they actually do something. Others lack confidence so they need a bit more prodding. But while some students sit inert because they lack confidence or are unsure what to do, other students do nothing because they are seemingly apathetic.

It got me thinking that this experience is one that really ought to be formally evaluated. This was real math. A timed individual test is not a place typically for creative thought and does not allow enough time usually for one to start down a path, come to a dead end, and go back and start anew. But, the students today had ample time to play with the numbers, reflect on answers, think about what information they needed and the questions they would have to ask to get that information. Groups that were most effective had the best communication and students asked each other for help. The most dysfunctional groups sat back and hoped that one member would just do all the work for them.

Perhaps, I thought, I should make this a graded exercise after all. Would that be so unfair? Why not create learning opportunities without deciding ahead of time whether or not to make it a graded exercise? Why not wait and see if the exercise is a good evaluation tool before committing to making it graded? Sort of like a reverse pop-quiz. Surprise! I am going to grade what you just did.

In my eyes, today's activity would have provided a very authentic evaluation of a student's commitment to learning and of their problem-solving skills. But, I did not feel it fair to do this as I had never told the students ahead of time that this might happen. I think I could have justified it, but I did not want to deal with the predictable complaints from the students.

But, as I write, this sounds like something I may try going forward.

Would students feel that they were constantly being judged? Would that be a bad thing?

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...

Weighting grades

The following is a true story.

I am an advisor at my school. There are five students who make up my advisee group. One of my primary functions as Advisor is to monitor their academic progress (aka grades) and communicate to their parents how the advisees are doing. This means I send emails every two weeks to my advisee parents in which I write their current grades and my spin on those grades. You know, progress reports. Mostly, the parents care about the grades.

One of my advisees, let's call her Chris, is in AP Chemistry. Her parents have emailed me all of two times in the year and a half that I have been her advisor. So, you can see they are not the most hands-on parents. Thus, when Chris came to me with a question about her AP Chem grade, I figured it was from her, and not from her parents. Before Christmas break, Chris had a B+ in AP Chem. Since that report, Chris had earned an A for Participation and an A on the alternative assessment (a substitute for a final exams which were cancelled due to school being closed early... but that's a different story). So, essentially, she had a B+ in the class, and then two more grades were added to her AP Chem gradebook that were both higher. Her final grade for the semester? B.

So, Chris emailed me and asked me about this. I had not yet had a chance to notice the change in grades. I was happy she brought it to my attention. I looked into it and discovered that her grade had gone down despite her last grades being better. The reason? When the teacher had initially set up his computer gradebook, she had not set it up properly, or it had been changed on her, and the weighting was off. So, she had readjusted her weighting and that had changed grades a lot and resulted in Chris's grade dropping.

Do you know what a weighted average is? Google it. Teachers use it all the time. A grade may be on a 100-point scale, but those points are not distributed equally among the different kinds of assessments. Teachers usually calculate averages for each category, and then weight them according to their significance. For example, tests could be worth 60%, quizzes worth 25%, and homework worth 15%.

Anyway, the AP Chem teacher, let's call her Amy, weighted her assignments this way: tests 25%, quizzes 25%, labs 25%, homework 20%, and participation 5%. On the surface, there is nothing terrible with this distribution of weights. It is perhaps odd that tests and quizzes have the same weight, but I can imagine how this might be appropriate. Dig a little deeper, and an interesting fact emerges: there were only two quizzes while there were four tests. This means that each quiz was worth 12.5% (25% divided among two quizzes) of the course grade while each test was worth 6.25% (25% divided among four tests). Or, to put it another way, the teacher's weighting meant that a quiz was worth twice as much as a test in the course grade.

Now, I did not know how Amy viewed tests and quizzes for her AP Chem class, so I asked Chris about what her AP Chem teacher had said regarding the importance of quizzes. Chris remembered that her teacher had said quizzes were worth less than tests. Unfortunately, Amy's weighting did not make that a reality in the calculation of the course average.

I sat down with Chris and we looked at her averages and calculated her course average using a variety of weight distributions. In the end, it did not change Chris's letter grade. It certainly changed her numeric average, but it did not take her from a B to a B+.

However, it would be very possible that a change in the weighting could change the letter grade of a student whose numeric course average was on the cusp of the next letter grade.

For example, suppose Cory had a 92 test average, an 84 quiz average, a 90 lab average, a 90 homework average, and a 93 for participation. Using Amy's weights, Cory's numeric course average would be 89.2, which would be a B+. If the weighting were changed so that tests were 40% and quizzes were 10%, then Cory's average rises more than a full point to 90.4, an A-.

For those of you who know, the difference between a B+ and an A- to the University of California is huge. A B+ gets a score of three while an A- gets a score of four in Grade Point Average (GPA) calculations. Thus, this teacher's seemingly arbitrary designation of grade weights could have a very real consequence to a student in the college admissions process.

Do you still believe in grades?

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.

This is a test. This is only a test.

Grades. Yuck. I don't like them. My students don't like them. Alfie Kohn doesn't like them.

Here I am sitting in my classroom while my Algebra 2 students are taking a "unit test". Why? Because I am supposed to assign each student a letter grade that is meant to represent their... what? Math ability? Feats of memorization? The ability to regurgitate what I told them? Their understanding of certain math problems on this particular day?

A test is a pretty easy way for a teacher to defend a letter grade. Compared to other forms of assessment, they do not take too long to create, correct, and grade. Math tests in particular tend to be easier since there is (normally) one correct answer and only a few different ways to arrive at that answer. It will probably take me two hours of work to correct and grade the test my students are taking right now.

But how valuable is it really? And I am in fact harming the love of learning that I want to instill in my students by giving this test? And if I think that I am, why on earth don't I stand up and say, "Stop! Put your pencils down and let's get something meaningful done."?

One, because I'm tired. Last night, when I sat down to write the test, I had a pretty good idea how long it would take me and by what time I would be able to let my head hit the pillow. As I said before, tests are easy. I want to spend time with my two-year-old and with my wife and I know how to write a test and how evaluate a test. It's a known quantity and is not scary for me. I feel pretty certain that after I finish this test, it will be a decent tool to help me assign semester grades to my students.

Two, I prefer to follow the paths of those who have gone before me. And all of my colleagues from all of my schools, as far as I know, depend on tests to determine grades as much, if not more, than I do. Frankly, it's a bit daunting to jump off the diving board if I've never seen someone else do it before me. If I try something totally new, something that no one I know has tried, then I run a greater risk of the experience not being meaningful or effective. I only have so many teaching days and if I lose a week, I don't get it back. If I know someone has tried a particular project, then I feel more confident. I have been searching for alternatives to tests, but so far, I lack the necessary gurus. And don't send me a web link. A web link is not going to help. I need someone to sit down with me, and explain what they did and how it worked and why they liked it.

Three, I don't know that my school would look very favorably on me not giving tests. I do that with my Advanced Mathematics class, but that is not a pre-AP class. In fact, the typical student in there is likely never taking a math class again. So nobody can tell me that tests are necessary to get them ready for some big standardized test to come. I believe, but am not sure, that if I did not give tests in my Algebra 2 class, that I would get a "talking to" by some one. I know for certain that I must give a final exam (which I was planning to write during this period, but, hey, this blog is better for my soul, right?). It may be that I have sufficient flexibility in the format of that exam that I could design something that is not so test-like. But, I have a fear that would not go over so well. Hopefully, my fears are irrational.

I am sure there are other reasons I continue to give tests. But, I don't want my posts to be too long. So, now I shall stop. The testing, however, will continue...

The Scoreboard

I recently came across this table. It is a table generated by a secondary school that the school sends to all colleges and universities to which the school's students apply. These are the cumulative GPAs for the Class of 2012 at this school. This is an actual thing, that is, I did not make this up.

Why would a secondary school do this? Does this advantage students? What might a college admissions officer make of this?

What about the students? This is accessible on the school's web site (that's where I got it). If I'm a 12th grader at this school, then I'm in that list somewhere. What does that make me think? Is school a competition? Is this the scoreboard? Am I a winner? A loser?

I have no idea if other schools do this, but my guess would be that they do.

What do you think?

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...

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.

If not a letter grade... then what?

The basic theme of this blog is my disdain for the letter grade. As a professional educator, evaluation of students is a big part of my life. So, if I were to not use letter grades, then how would I evaluate my students?

The first idea that comes to mind is a narrative comment. Allow me to consider a hypothetical situation to better explain what I mean. I teach math, so students take many tests (and no, I do not like tests either, but that seems to be a discussion for a different blog... at the very best it's a tangent, and being a math teacher, I should know all about tangents, both the mathematical kind as well as those meandering ones such as when I start going off about tangents when I am supposed to be considering evaluation... )

Anyway, let us pretend that I give my ninth graders a test and the test includes problems on algebra, some on trigonometry, a few on matrices, some graphing, and some transformation problems. As this is math, I can create an answer key and I can correct a student's work against this answer key and I can quickly see a student's mistakes. I do not merely consider a student's answers; I also look at the supporting work, that is, the steps a student took to find, for example, the inverse of a matrix or to solve an equation. So, regardless of a student's answer, I can see the methodology and, really, the thinking a student utilized to arrive at their answer. Thus, I can evaluate not only the correctness of their answers but also the appropriateness of their method and their ability to communicate this method.

So, I go through and correct their test. I mark which answers are right, which are wrong, and I indicate where their method is appropriate and where it is not. Like most math teachers I know, I can award points for correct answers as well as for supporting work. So, then I add up all of these points and arrive a number. Now, many math teachers would know the possible number of points one might earn on a test, and at this point they would divide the number a student earned by the total possible to get a percentage which they would then convert magically into a letter. (See my last post to read some of my opinions on that.) But, the point of this post is to consider what to use in place of such a letter grade.

Back to the student's test... they have earned a number of points. Some of these points are for correct answers and some are for correct methods. Now, that number of points itself is feedback for a student and is thus a form of evaluation. Moreover, if I indicate clearly where they earned points and where they did not, then that is another level of evaluation. A student can see if they used the proper method and if their use of that method led to a correct answer. That, much more than a letter grade, tells a student if they have correctly learned how to find the inverse of a matrix or if their equation solving techniques are strong.

So, without any narrative comment (and without a silly letter), I have already provided a student with copious information regarding how they did. But, I have not really passed any sort of judgment yet as to my opinion on the quality of their work.

So, now the narrative comment becomes useful. I could convey to a student if I think they did well or not. I could judge their performance compared to earlier work. I could specifically state if they were strong on matrices, but weak on algebraic equations. I could clearly point out whether they chose the best methods or not and how accurate their answers were. I could also judge if they have been learning as much as I think they should have up to that point. Additionally, I could outline areas that they ought to focus on as they strive to improve.

Such a narrative could fill half a standard sheet of paper or more. Such a narrative could take 10-15 minutes to write. Such a narrative would provide far more useful feedback to a student than any letter grade could ever hope to.

But, what is the cost of such a narrative, filled as it would be entirely with my opinions? Does my opinion on the quality of the work really matter? Couldn't my corrections stand alone as all the evaluation I do and couldn't I then let a student make up their own opinions on the quality of their performance? That is a discussion for a future post.

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?