Grading Papers

Without exception, retired teachers I have spoken to agree that the best part of retirement is not having to grade papers, especially on Sunday night. And yet there was something very satisfying and rewarding about looking at student work in detail. Grading a student's paper was like spending a few minutes inside of that student's mind. It took a lot of time and it was hard, but carefully grading papers was an important part of the teaching and learning experience.

My last blog was about how grades influence the learning experience. This one is about how the details of arriving at one of those grades are equally important. Papers must be graded fairly. A graded paper should contribute to a child's education in a meaningful way, regardless of the grade attached to it. I would like to share some of my thoughts about the process of grading student work.

At the top of every paper I intend to grade is the sentence :

"You must show enough work so that I can reproduce your results."

I have found that this phrase solves a lot of practical problems about how much work a student needs to show. It also helps when the students solves a problem in an unexpected way. It allows a student to use technology intelligently as long as I am given enough information so that I can get the same result using the same technology. It enables me to effectivly evaluate the error a student has made and give the appropriate amount of credit for the work.

This leads to another aspect of grading a student's work that developed carefully over the thousands of problems I graded. I give points for correct mathematics. I do not take off points. When a student looks at the number of points earned for a particular problem, the student will see a +10, not a -2. The students got the 10 for doing several things correctly that would have lead to the correct answer. Unfortunately, the student made an arithmetic error when computing part of the answer and so did not earn the 2 points allotted for determining the correct answer. It should be noted that one consequence of this grading policy is that a bald answer without supporting work will get 2 of 12 possible points.

We are teaching mathematics. We are assessing the quality of mathematical reasoning a students is capable of. That means we need to see the process used to arrive at the solution, and it is the process we are evaluating. It is inappropriate and short-sighted to require the students to use the process we expect, but we cannot evaluate what we can't see. So, if a student uses a guess-and-check method, I need to see the guesses and the checks. If the student uses intuition and evaluation, I need to have the intuition explained, and I need to see the evaluation.

Consequently, it takes a long time to grade a set of tests. The effects, however, make it worth the effort. I have learned a lot of mathematics by following the work of a student who took an unexpected path. But more importantly, grading lots of papers teaches the grader what sort of misunderstandings students have, and that in turn enables the grader to try to find ways to eliminate thsoe errors next time around. At the very least, one learns to warn students about a mistake students often make on a particular kind of problem. At the most, the teacher can modify the problems used to teach the concept so the class will make the mistakes early on, exposing and hopefully eliminating the potential for those mistakes to occur.

Another consequence of grading this way is that the grader learns a lot about the particular habits of mind of the student. This information in turn is very helpful when the student or the parent wishes to know what can be done to improve. The teacher is aware of a lack of organization, or not checking work,or poor computation skills, or careless marking of a diagram or any of the other habits that interfere with success and can communicate those habits to the students and parent.

Comments to the student congratulating a clever move will have more impact than criticism about a bad move. A teacher can take the opportunity to point out specifically to a student what might have resolved the error. The feedback is up-close and personal and has impact.

But perhaps the most important aspect of grading papers this way is that it conveys a sense of value to students. It implicitly tells them that the important part of mathematical work is the process. They will get points for failed attempts if those attempts are appropriate and reasonable. They will get more points for a clever observation than for remembering a few steps from a previous problem. The students will learn that mathematics is about logical reasoning and making connections more than mathematics is about remembering rules and following them carefully. And that is a big deal.

Grades

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.

"I think I discovered a theorem, Mr. Karafiol!"

This past week, one of my students stopped me on my first pass around the class with the exciting words "I think I discovered a theorem!  In a right triangle, the angles the median makes with the hypotenuse are twice the other angles!"  We had discussed the median to the hypotenuse in the previous class, but I'd never thought about the angles before.  So I started doing algebra in my head, confirming the relationship he was describing:

Before I could get very far, Vik showed me the elegant diagram below:

With the inscribed angle theorem, the proof is immediate.

Eureka!

Low Tech, High Impact

Although my department has a reputation for integrating all kinds of whiz-bang, high-tech devices into our classes, we've become increasingly excited by one decidedly low-tech tool:  large whiteboards.

The model is one I learned from our AP Physics teachers several years ago.  Contrary to my usual one-problem-at-a-time practice, I give students a handout with several problems, and instruct them to work on one that they find tough--so that they can build their mental muscles.  Then, each group gets a large whiteboard, and writes up a good solution.  Groups present their solutions and field questions from the rest of the class.

When I first heard about this technique, I thought it was a poor second-best to having every student work on every problem.  But I tried it, and now I've seen some substantial benefits

1. Students get to see a variety of problems worked out carefully, and ask questions in a nonthreatening setting.  And while they're not grappling with each individual problem, the more questions they ask and answer, the more they think through the problems presented--even ones they didn't work on.
2. Students work on communication skills.  In other instructional models, it's rare to have an entire student response of this length subjected to a full-class critique.
3. Students focus on one problem for enough time that they can think deeply about it, and transfer the ideas to other contexts.

The class above is my AP Calculus AB class, working on optimization problems.  After a brief introduction, groups chose between four:  minimizing the distance from a point on a line to a given point not on the line, minimizing the surface area of a can with fixed volume, maximizing the area of a rectangular pen with dividers, and the traditional "box problem" shown above.  In this case, I had two major goals:

1. Students needed to understand and be able to recreate the major steps in solving any calculus optimization problem: identify the constraints and the objective function, rewrite the objective function using a single variable, and apply calculus to find the maximum or minimum value.
2. Students needed exposure to standard problems, both in case they should encounter those very same problems later, and to provide foundational experiences from which they could build out to other, similar but perhaps more-complicated situations.

Whiteboarding was perfect: it was enough for kids to see and discuss the different setups--some of which they had encountered in precalculus--without having to struggle through creating each one themselves.  (Assessment today: they were pretty comfortable doing another distance minimization with a harder graph.  So it worked!)  And the in-depth thought about the process really paid off: again, in today's followup class, students seemed able to articulate the major steps and follow them with little confusion about sequence or "what comes next?" kinds of questions.  Finally, their writing and communication were much stronger than I would have anticipated:  students presenting today wrote clearly and identified major steps and results along the way, and other students used vocabulary correctly without hesitation.

So there you have it: for less than $15 per board all-in, a mechanism to facilitate thoughtful collaboration, presentation, and discussion. 

Postscript: if you're looking to acquire some, we get ours from whiteboardsusa.com--they ship quickly and have reasonable prices.

Homework and more homework.

A year and a half ago, two other experienced teachers and I gave a workshop about teaching geometry. Many of the participants wanted to know how we handled homework. We wanted to have them do geometry activities but decided to allot some time to homework since that was their concern. The other presenters and I decided that the three of us would each get a minute to explain how we managed homework. We did not consult each other beforehand but the three of us were on the same page about virtually everything regarding teaching and learning geoemtry, so we assumed we would support each other on this issue.

It would be hard to find three more different responses than the three we gave. Our descriptions of how we handle homework make the Tea Party and the current administration look like they are in agreement on almost every issue. Each of us had arrived at very different conclusions based on our years of experience.

Having said that, I think I will contribute my thoughts on the subject. I tried a lot of things and all approaches had flaws. I eventually came to several conclusions about homework:

1. Homework is necessary. Students need time, on their own, to wrestle with mathematical ideas, to put them into perspective, connect them to other ideas and to practice. Some of this can happen in a class, but it is the rare students indeed who is able to make connections and internalize concepts in one class period.
2. It needs to be done on a regular basis, so students come to class knowing more than they did when they left class the day before. Otherswise there in not much progress happening.
3. It needs to involve reading and learning from reading. A students who can not read text, study an example, and determine meaning, is not an independent learner and will be forever limited in his or her ablilty to further advance his or her education.
4. Students need incentives to do homework in mathematics even though they know it is helpful and important. They are human and they are immature. Given the choice, they will usually do something that is more social or more to their own interests than doing the problems that the teacher requires that they do. Even my best students, when homework was not required, spent their time doing the Chem assignment or writing the history paper that was due tomorrow, instead of the math, if they could put it off for another day. "Never put off until tomorrow what you can just as well do the day after tomorrow." --Mark Twain
5. Homework is a learning experience, not an assessment activity. As such, it should not be graded. It should be evaluated in proportion to the effort that went into it, not in proportion to the number of correct answers.
6. Students need feedback with regard to the correctness of their work.
7. My time, both in and out of class, is better spent writing interesting problems, asking good questions, and writing appropriate assessments than it is spent grading homework.

So, after many iterations, I devised a plan that worked tolerably well for me. Here it is:

When students enter the class, they are provided with solutions to the homework that is due. Sometimes this was done by displaying them on the screen in the front of the room. sometimes it was done by placing copies of worked-out solutions in their mailboxes so they could retrieve them as they entered the room. (Every student had a mailbox so that I did not have to waste tome passing out papers.) Were I still teaching, I might resort to putting the solutions on my web page at an appropriate time. At any rate, students always had a chance to see what I thought was an appropriate way to solve the problems. These were more than answers; they were solutions.

I also gavemy students a problem to work on during the first few minutes of class. When class started, I walked around and looked at the homework to see if it looked appropriate. If so, I gave them 2 points. If not, I told them to finish it in a proper manner for tomorrow. If it was turned in the day after it was due, they got 1 point. Otherwise, 0 points.

As I walked around, my students had a chance to ask me about a particular problem or two that they were still confused about. I would make a judgement based on this information about the appropriateness of working a homework problem with the class. Usually we did not.

We were on a quarter system, about nine weeks per quarter. At the end of the quarter, any students who had done all of the homework on time (or at most missed one or had one or two late--that is, 2 points off of the maximum) had earned THE HOMEWORK BONUS! That meant the their lowest quarter test grade would be raised by a grade. I graded by letter grades, not points, so that meant that, for instance, a C became a B.

It worked for me with my students. Most students did their homework daily and were very concerned about not losing that bonus.

Oh, there was a dark side. If a student missed more than a week's work, five assignments, their highest test grade was lowered by a full grade for every one they missed after five.

I had wanted the homework score to be entirely positive: If you do this, it will advance your work by a bit. I found that after students lost the bonus, some decided it was no longer necessary to do any homework, so I had to include the negative part. It worked as well. Students do not like losing something they have already earned.

I am not sure how much of the homework I looked at had been copied. I never did find a good way to combat that problem, other than giving two or three unannounced quizzes every week. All of the quizzes counted as one test. They were short and I thought of them as formative more than summative, but they did count. And I did learn quite a bit about what my students had learned and what they had not yet learned, based on these quizzes.

I think this system worked very well for me. I hope there is a part of this you can use as well.

Of Babies and Bathwater

Thanks, John, for those kind words.  And yes, I plan on presenting more proofs.

These last two weeks have been busy with end-of-quarter grades, projects, etc.  (A forthcoming post will be "High-school level projects that involve actual mathematics," but I digress.)  This post is reflecting on the not-entirely-successful first iteration of our "no homework grade" policy: nightly homework assignments don't count towards students' grades, but frequent unannounced quizzes use representative homework problems as an incentive to complete assignments and an assessment of whether students know how to do the math.

The data are in and the following seem clear:

• Students are completing less homework.  How much less is unclear, because in the past, the marks on students' papers didn't always correspond to thoughtful effort expended on mathematical problems, but in my upper-level classes, it's typical for only about half the students to have attempted a significant number of problems, and in previous years, it was more like 80%.  In the past, I doubt that the typical homework assignment was copied/scribbled from answers by 30% of my students.  Lower-level classes are doing better on most days.
• Students perceive the policy as "you don't have to do homework," which seems like a misreading to me.  More accurate would be "homework isn't graded and factored into your overall grade." 
• Students are doing less well on in-class tests than they did on last year's tests, although results vary by class.  Classes that are giving a lot of homework quizzes are finding less dropoff, but those same classes have younger students.

In his fabulous book, So Much Reform, So Little Change, Charles Payne dissects the etiology of failed reform efforts, discovering that a pervasive symptom is simply discarding changes that don't work, rather than examining and adjusting them.  In that light, our department is trying to revise our new homework strategy rather than revert to one we found problematic.  Some questions and possible answers:

1.  Why don't students do homework under the new policy, when they can see their grades dropping?  First, they may not see the connection between doing homework thoughtfully and actually getting better at math: it's telling that large numbers of students describe the policy (to parents, counselors, and teachers) as "you don't have to do homework" rather than "homework is important to learning, but you're only graded on what you learn, not what homework you do."  Second, they have a lot of other work--we're suffering from being the "first movers" in responding to Race to Nowhere.  When a student is up at midnight and choosing whether to do math or go to bed, the threat of a possible homework quiz is clearly not enough incentive to do a handful of math problems.

2.  What can we do to improve the situation without throwing everything out?  First of all, we can communicate better about what we think it takes to learn serious mathematics.  Maybe we should take a cue from the advertising folks, and make posters saying "192 minutes is not enough" or comparing time spent doing math to time spent doing other valuable activities?  What if we change the homework quiz policy so that homework quizzes are frequent but expected, for example every Monday and Friday?  What if we specifically identify which homework problem serves as the basis for each quiz question?

3.  What other ways are there to get students to do math outside of class without increasing incentives to cheat or skate by?  Webassign?

This is a tough time for us:  we want to do things differently and better, so we need to figure out how to adjust our course rather than simply u-turning.  Any ideas and suggestions are welcome; we'll take them in the comments.

Presenting proofs 2 etc.

I applaud P.J. for working through the proof of Brahmagupta's formula with his advanced students, even though it was perhaps not in line with what we call "best practice," as many of his students may not have understood.

First and foremost, one of the most important things we can do for our students is to share our enthusiasm for mathematics, perhaps even mathematics they do not at this point completely understand. Knowing P.J. as I do, I'll bet his students were as excited by his excitement as they were by the beautiful proof he shared with them. I am sure he did not just stand at the board with his back to them writing. Probably didn't even use a board.

Secondly, it is critical that every student see some beautiful and significant mathematics. Part of being educated is understanding where knowledge comes from. How is it that we come to know the things we know? How did we come to think this way? How did anyone ever think of that?

Seeing a beautiful--if complex--proof is as important to an education as seeing a complicated, if not completely understood painting, perhaps a Picasso, or a DeKooning. Seeing a beautiful proof is as important as hearing music written by Schoenberg or played by Charlie Parker or Ornette Coleman, or attending a play by Beckett or Shakespeare. The proof or the painting or the music or the play may not seem beautiful the first several times we encounter it, but we are aware that creativity and beauty are present, and sometimes it takes us work and time to take it all in. When the complex, beautiful thing does make sense to us, we are changed. That is a part of education. Not the part that will necessarily make a lot of money, certainly not a part that will improve our test scores on a high-stakes test, but a part that will improve the quality of our lives.

Mathematics is overflowing with creative ideas and contributions from creative people. People like Cantor, Godel, Gauss, and Newton had some amazing ideas and made some remarkable contributions. Mathematics teachers have a responsibility to make our students aware of the inventive nature of mathematics, and it is easiest to do so if we share the problems and proofs we love. I used to teach a unit on non-Euclidean geometry just because I liked the ideas so much. A few of my students grasped what I was telling them and studied it further. I suspect others did much later in their lives, and I just have never heard.

I recall my Modern Algebra professor, Dr. Pilgrim, comparing a proof he had just showed us to a Gail Sayers run he has witnessed the Sunday before during a Bear's victory. His comment stayed with me and made me take a harder look at proofs. As it happens, I have made a list of my favorite mathematics. In the top ten, seven are proofs. I didn't always feel that way, but then I didn't always love mathematics the way I love it now. Mathematics has clearly made my life more interesting and the opportunity to share that with other people has been even better.

I rarely share those proofs with all of my students because I know that the timing has to be right in order to have in impact. Many students shut down as soon as they see a proof coming. It is such a shame that they are missing out on such enjoyment. But then a lot of people don't listen to jazz, classical music, go to art museums or serious plays either. I find all of those interesting and fulfilling. I am sure there are things I am missing that are equally important, but no one ever hooked me on them. Such is the way of the world. All I can do—and I must do it—is to try to share with others those things that bring me such joy and hope some of it will rub off, so we can share it together, and so they can keep it going.

This weekend, I attended the Illinois Council of Mathematics Teacher's annual conference. One session I attended was organized by Doug ORourke, a good friend of PJ's and of mine. Coincidentally, part of what he offered was an opportunity to investigate Hero's formula and Brahmagupta's formula in a new and different way. He proposed several versions of what could have been the formula and challenged us to find a way to explain why each variation of the formula could not be correct. The discussion was stimulating and resembled the discussion that comes before a proof and rarely happens in any math class. He then took Brahmagupta, had us enter it into a CAS calculator and then enter all sorts of numbers, to see what would happen. The overriding theme was Plausibility. By this he meant to look at special cases, impossible cases. What a brilliant way to spend an hour. Thanks Doug and I shall take this with me.

Friday evening was devoted to awards. The outstanding secondary School teacher award went to Natalie Jakucyn, truly a giant among us. In her acceptance speech, she thanked her high school math teacher, a nun who held her students to very high standards. Natalie recalled the day Sister put a long proof on the board. When she was done, the Sister wrote, "QED," went the back of the room, and said "Isn't that beautiful?" It was then that Natalie decided to become a math teacher!

So, thanks again, P.J., for taking the time out from the usual hands-on, engaging, student-discovery type of lesson that is typical of your classes and inspiring at least a few of your students by showing them a proof. Do it again. No one should graduate from high school without seeing Euclid's proof that there are infinitely many primes. It is certainly a proof that is in "The Book." and has inspired many a fledgling mathematician. And an important part of excellent teaching is inspiration.