Wednesday, January 11, 2012


Grades are the elephant in the room when it comes to learning and teaching. They are an ever-present part of the relationship between teacher and student in most educational situations. Grades can interfere with learning if students believe that no matter how hard they work, their grade will not improve, or if students believe that they can get an A without doing much work. Grades can destroy that partnership if students think the teacher is unfair. Grades consume a large percentage of a teachers time.

Perhaps the worst part is that frequently grades become the goal and overshadow learning. Teachers sometimes use grades to coerce students to comply. Teachers do things like not giving credit for math work done in pen, even if the work is exemplary. Students use grades as an excuse for not doing work. Students will not investigate a problem further bacause they know it will not be tested.

And yet, grades persist. I think they can play an important role in education. Students and parents need feedback about their accomplishments, they need advice about how to improve, and they benefit from legitimate praise and criticism.

Allow me to share two very personal experiences that are vivid memories of my grade school days. For two years in a row, I was told by the music teacher that I could not sing. (She was right.) I was then instructed to mouth the words as the other children sang. As I grew up, I realized that I love music, but I can't sing, so I don't try. I have been told by several people that anyone can learn to sing. I don't believe them, because I was told at a very young age that if I sang, it would interfere with the singing process in class. I can't do singing. The second experience happened in eighth grade when my social studies teacher was trying to explain inflation. I rasied my hand and asked a question about the consequences of inflation. He looked at me and said, "You must be really good at math." Fifty years later, I remember that moment in class; I have devoted my life's work to mathematics.

Both teachers were assessing my work. Neither assessment had anything to do with grades. My grades in school were never very good because I frequently didn't comply with the teacher's wishes about how to do the work (I really liked doing math with a pen, for example). But I did learn and so consider myself to have had a good education in spite of all those C grades.

When I started teaching, I had the good fortune to be in situations where I had the freedom to decide how grades were going to be given. I spent a lot of time thinking about it, tried many plans, and eventually hit on one or two that worked. I think my grading schemes helped me become a better teacher. I would like to share some of my thoughts about grades in the next few blogs.

I have always thought that in order to earn a good grade, a student should demonstrate knowledge of the subject and the ability to apply that knowledge in a variety of situations. That means that each assessment should include some routine exercises to see if the student has learned the basic material, problems right out of the book with different numbers. The students should also be expected to do problems similar to some of the really hard problems that we did in class. And the student who wishes to earn an A ought to demonstrate the ability to apply the information from the unit, as well as from the entire course studied so far, to a new situation. So, my tests are usually about 50% routine problems, 25% difficult problems that are similar to problems they have worked on , and 25% original problems. I give them one period to work on the problems unless they are legally entitled to more time. I carefully look at their work and give credit for correct mathematics relevent to the problem.

Solving a difficult problem takes time, and there is often a certain amount of luck involved. A promising approach may lead to a dead end through no fault of the problem solver, while an equally promising approach may work just right. If we intend to assess out students' success as problem-solvers, we must ask them to solve problems on tests, not just do exercises. That in turn influences how we associate a grade with work done.

I would like to know who decided that 95% was the benchmark for excellent work and what sort of work were they thinking of. Nothing I can think of that is reasonably difficult can be done correctly 95% of the time. The best baseball players that ever lived were successful at getting on base if they could get on base 40% of the time. Most players don't even come close, because hitting a baseball is very hard to do. A 30% success-rate is outstanding.

One of the national standards of excellence in the U.S., the Advanced Placement test, gives only five grades: 5,4,3,2, or 1. In order to get a 5, a student needs to get approximately 72% of the test correct. That level of excellence will often earn college credit for the course in question.

A score of 100 or more out of 150 is considered outstanding on the National Mathematics Exam offered by the Mathematical Association of America every year. That score qualifies a student to move to the next level of competition and often means that the student was in the top 1% of students taking the exam.

P.J reminded me about Dr. Paul Sally's rubric: "If you're getting 50%, you're doing well." Dr. Sally taught Honors Analysis at the University of Chicago. No one ever accused Dr. Sally of having low standards.

That brings me to another thought about grades. My first two years I computed grades two ways. I kept track of total points earned by students, and I also assigned a letter grade to each assessment and then used the letter grades to determine a final grade. It became apparent theat the letter grade method was far superior in two respects. First, students always seemed to have a feeling for where they were. Second and more important, the letter system was fairer, the letter system meant the grade was less influenced by a really bad test, and the letter system allowed me to assign points to problems without regard to making the total come out to a pre-specified number.

Another principle I followed without exception: Every problem was worth the same number of points. I didn't want students to have to worry about how much time to spend on this one or that one bacause one problem was worth more points than another. I did want my students to look the problems over and work the ones they were most confident about first. Since I established the cutoff points, this was a very good and fair policy.

The point is that the elephant is there, and it matters. Grades influence our effectiveness as teachers, and we must spend considerable time and effort working out systems that enhance our teaching, emphasize the things we think are important, inform parents and students about the quality of their work, and are even-handed and fair.

Next time I will share with you some specific things that worked for me. Until then, please reflect on the grading policy that you are using and how it alters your ability to teach mathematics. No matter what you think, it does make a difference.

More later.