Chapter 14 - Writing about Grading

Grades are a topic of great interest to students. Some of the most frequent questions I hear in class are questions about grading: "How much is this assignment worth?" (I sometimes ask in return, "how much to you think it should be worth?"), "Will this be on the test?" (I always answer "yes", or they may stop listening!), and "What's my grade?" So I have decided to get them more involved in this issue. One project related to this important issue is the one they dubbed "Think Like a Teacher". I gave each student a worksheet with a hypothetical test question and 4 student responses gleaned from my collection of common errors that students make.

This project was one of the most interesting, at least to the teacher! Teachers spend a lot of time grading, but in many math classes, tests may be multiple choice, true/false, or questions with short numerical answers. Students often ask if they can earn partial credit for their work, rather than be graded on a right or wrong basis, as is often the case with numerical problems.

In this project, I asked the students to grade some examples of a fictional student's work. The question at issue was: "Triangle ABC is inscribed in circle O. Angle C is a right angle, and AB = 6. Find the area of the region inside the circle but outside the triangle. Draw and label a diagram and show all your work." I had prepared 4 fictional responses, all of which included a drawing and some equations, calculations, and answers written by the fictional students. These responses, with diagrams, are shown below. In the first example the fictional student had misinterpreted the question and drawn an equilateral triangle, shown all work correctly for that diagram, but had written an incorrect answer. In the second response, the student had drawn a correct diagram and shown all their work but made a small numerical error, thereby coming up with an incorrect answer. In the third response, the student had drawn an incorrect drawing, showed no work, and had the correct answer. In the fourth response, the student had drawn a correct drawing, showed all the work, but found the area of the region inside the triangle rather than the region that was requested.

 

The instruction to my students was to grade the 4 fictional students' work. This was a group project, with 3 or 4 students in a group. First, the students in each group were to do the problem themselves, and agree on the answer. Then they were to grade the work done by the fictional students. They needed to decide if partial credit was to be given, and if so, on what basis? If the problem is worth 4 points, would they give points for the drawing? Would they give partial credit for writing the correct formula even if the mathematical portion of the solution was incorrect? They were also asked to explain why they chose to grade in the way that they did. In addition, at the end of this assignment, I asked them to write their thoughts about grading in general. I asked some open-ended questions: "What is fair? Which do we value more, the answer or the method? If someone drew an accurate diagram, labeled it correctly, wrote the correct formula but made a small error in arithmetic, would they receive any credit? What if a student showed no work at all, but wrote the correct answer?"

There was much conversation, heated discussion, and lengthy responses. This seemed to be a topic dear to their hearts; no one had ever asked their opinion on this issue, at least not in a mathematics class. One group named this project "Think like a Teacher". In their reflections on this assignment, students made many thoughtful and interesting comments. For example: "I particularly enjoyed this writing assignment because it wasn't a typical math assignment. Instead I explored what teachers must go through when they grade tests and papers. The criteria for such grading is very difficult; I found I had to choose what I valued from the so-called students and what was fair. Also, I received some good practice for the next test because I was able to see what mistakes to avoid." Bennett

Another said: "The task seemed simple enough, but I was surprised at how difficult it really was! I tried to be fair and grade justly, but it was difficult to do so. Some of my teachers have given me partial credit for showing my work on tests, although I got the answer wrong. I really appreciated this because I felt it stressed the importance of understanding the concepts and not just the accuracy of computations." Sophie

Sometimes I assigned this project as a group project, with three of four students in each group. Occasionally, I asked each student to work alone. It seemed to promote deeper thinking when students worked together on it; they found that each member of the group had differing opinions, and the conversations that ensued were as valuable as the essays that they wrote on the topic. Richard and Joshua, a group of two, had this to say: "We based our grading on three assumptions. One, that if you got the right answer, you probably were able to apply the right method to get that answer. Two, if the method is written down correctly, it shows that the student knew what he or she was doing, even if the answer ended up incorrect. Three, the benefit of the doubt should be given because we're all human. The point system we used is based on giving 5 points for the each problem. These 5 points in this problem are 1) finding the area of the circle 2) finding the area of the triangle 3) applying the concepts of the radius 4) knowing about how a right triangle fits into a circle 5) and last, putting it altogether.

This group went on to say, in their final detailed analysis:

1) First we'll grade Ann. She demonstrates a fairly clear understanding of the concepts of the radius of a circle and a 30-60-90 triangle in a circle through the diagram. 2 points. She also shows she knows that the area of the shaded region is found by subtracting the area of the triangle from the area of the circle. Another point for getting the area of the triangle. However she shows no written equations or how she found the area of the circle (which is incorrect, anyway) and how she found the area of the triangle. Therefore, we decided to give her a 3; she didn't get the area of the circle right and didn't show the intermediate steps.

2) Then there's Ben. He doesn't understand the concept of a right triangle inscribed in a circle. But he shows his work and knows how to get the area of the triangle and the circle. He doesn't know the concept of the radius though, and labels it as 2 root 3 although the diameter is given as 6. Because it is a right triangle inscribed in a circle, its hypotenuse was supposed to lay on the diameter. Therefore, we gave him only 2 points.

3) Cara had the correct answer, but her diagram showed that she didn't understand that the hypotenuse is supposed to lay on the diameter. But she did label the point and the 90 and 30 degree angles. She also was unclear how she figured AB was 6. So we gave her 3 points.

4) Dan showed he knew the concepts of area of a triangle, and area of a circle. He also knew the concept of hypotenuse on diameter for right triangles, as well as the radius. The reason he got it wrong was that he probably thought it was a 45-45-90 triangle, showing that he had not labeled it properly. Also missing the labeling of the points. However, because it was more complete that the others in understanding, we gave him 4 points.

This group went on to say: "Grading a problem more heavily on the correct answer is less fair than weighing it by the demonstration of methods, work, and diagrams. In some ways, knowing how to do it is more important than just getting a right answer, because sometimes you can get the right answer just by guessing!"

Chanel wrote a rather lengthy essay on this topic, in which she expressed her frustrations with traditional methods, and her preference for a grading system that gave credit for correct ideas even if there were some small arithmetic mistakes. In her essay, she said: "Math is a really hard subject. It is so exhaustingly exact and no mercy is shown. There don't seem to be any grey areas; it's all white or black, right or wrong. However, in this assignment, I saw a glimmer of hope - a possibility that we might be allowed to bend the unbendable rules, just this once! I feel that the method used, and the steps taken, can be as important as the precise mathematical answer. While the precise answer is the pearl of mathematics, the method is the very oyster that renders it. While a pearl is much admired, the oyster that produced it is cast off. Such should not be the case in teaching and in learning. I feel that the thought process in solving a mathematical problem is just as valuable and important as the answer. And the paths that you may venture down in your mind are worthwhile even if you do go astray, with a numerical error or inaccuracy."

I believe that this is true, and although I did not grade every assignment in this way, I tried to include partial credit on many problems, both in daily work and on tests. There are sometimes gray areas; and the teacher may find it hard to decide exactly where to give partial credit for things such as the correct method but wrong answer. But the students really tried very hard to show all their work and were not only grateful when given "part credit", they were much more careful with writing clearly and following sequential steps that their careless errors decreased, and their overall work improved measurably. From the teacher's point of view, the extra time spent in reading their work and in grading it was definitely worth the improvement in both achievement and morale!

My students seemed to feel this way too. In a group reflection on this project, Matt and Ross wrote the following:

"The aim of the instructor in any situation is to be able to evaluate the students' understanding as accurately as possible. Unfortunately it is not really possible if the questions require "yes/no" or multiple choice answers. In a perfect, fair world, there would be some way to get inseide the minds of students to flawlessly gauge their understanding of the subject.

Teachers, it must be understood, can never be really fair in every way. They can come close, but there is always a barrier that exists between what the student thinks and the teacher's impression of what the student thinks. The most fair method, in grading, we believe, is to evaluate the work not just on the mathematical right answer, but on the teacher's understanding of the student's thought process. Ben, for instance, got an answer that was nowhere near the correct answer. Was he wrong, then? In a sense, yes, and he should be punished for it. But what about the understanding he demonstrated in solving the problem he set up for himself? He shows a command of the core concepts of area formulas and the properties of 30-60 90 triangles that would satisfy any teacher. In this sensek he is right and deserves this recognition.

As is noted, it would be ideal if we could truly get into students' minds, thus eliminating the need for work shown. Great minds don't always think alike, yet the best way to pinpoint errors and examine pupils mastery of the subject is to request that work be shown. So, is method or answer more important? The answer is more important only insofar as it reflects sound understanding. At the same time, a cheater should not be rewarded for his skill at procuring other's answers! As the four problems show, each case must be judged individually with both method and result taken into consideration."

The instructions to my students were to grade the 4 fictional students' work. This was a group project, with 3 or 4 students in a group. First, the students in each group were to do the problem themselves, and agree on the answer. Then they were to grade the work done by the fictional students. They needed to decide if partial credit was to be given, and if so, on what basis? If the problem were worth 4 points, would they give points for the drawing? Would they give partial credit for writing the correct formula even if the mathematical portion of the solution was incorrect? They were also asked to explain why they chose to grade in the way that they did. In addition, at the end of this assignment, I asked them to write their thoughts about grading in general. I asked some open-ended questions: "What is fair? Which do we value more, the answer or the method? If someone drew an accurate diagram, labeled it correctly, wrote the correct formula but made a small arithmatic error, would they receive any credit? What if a student showed no work at all, but wrote the correct answer?"

There was much conversation, heated discussion, and lengthy responses. This seemed to be a topic dear to their hearts; no one had ever asked their opinion on this issue, at least not in a mathematics class. One group named this project "Think like a Teacher". In their reflections on this assignment, students made many thought-filled and interesting comments. "I particularly enjoyed this writing assignment because it wasn't a typical math assignment. Instead I explored what teachers must go through when they grade tests and papers. The criteria for such grading is very difficult; I found I had to choose what I valued from the so-called students and what was fair. Also, I received some good practice for the next test because I was able to see what mistakes to avoid." Jackie L

Another said: "The task seemed simple enough, but I was surprised at how difficult it really was! I tried to be fair and grade justly, but it was difficult to do so. Some of my teachers have given me partial credit for showing my work on tests, although I got the answer wrong. I really appreciated this because I felt it stressed the importance of understanding the concepts and not just the accuracy of computations." Kelly K.

I was amazed at the energy and thought the students put into their reflections on this process. They wrote much more than I had expected, and really put their hearts and minds into their essays. Some of the essays were written as a collaboration between 2 or 3 students; the following essay was written by Matt and Megan:

"In grading, there is no one absolute way to be fair, because different people value different things. Some people may value a correct answer over the method used, and the effort applied. In math, most people think there is only one answer to each question, and the answer is either right or wrong. But other people believe that the method used is a valuable part of the solution to a problem, and the thought process used to find the answer is as important as the numerical answer to the question. In this assignment, where we were playing the role of teacher, we chose to give the answer and the method equal value. We thought that the diagram was worth just as much as the steps in the solution, and the answer. Geometry is a visual subject, involving shapes, and if you are not able to construct an accurate diagram, then you cannot master the concepts. The work done to solve a problem is as important as the solution, because by showing your work, you communicate the process which you went through. By doing this, you and others can see what you did wrong or correctly. People who are able to describe accurately a method they have used to complete a task are better able to identify what went well (and why), what they want to improve (and how) and thus achieve a higher performance level. There is no one way that is better to solve a problem, as long as you reach the correct solution sooner or later. There are some methods that are more efficient than others, but to limit yourself to those would be to limit your thinking and options."

In grading another student's work, my class gained a great deal of insight not only in how teachers grade student work, but what they, themselves value, and how important it is to read the problem statement carefully, draw accurate diagrams, check your work, and the check it again! They found that seeing other people's mistakes helped them to avoid these mistakes themselves.

Their comments show thoughtful analysis of the task at hand, and an insightful appraisal of their own learning. As Megan said, "There is no one way that is better to solve a problem, as long as you reach the correct answer sooner or later. There are some methods that are more efficient than others, but to limit yourself to those would be to limit your thinking and options. However, if a method is not accurate (ie, making assumptions without proof to support them) then it is not a valuable method, even if you do reach the correct answer in the end." This comment shows a very thoughtful and considered approach to learning, and an excellent attitudue towards achievement.

Tiara wrote a fascinating essay on this topic; and essay that would have made any English teacher very happy. She wrote "I decided that the correct process of solving a problem is more than the correct answer." (I believe that this is true in the school setting; but not the best approach when designing an airplane! But the habits of mind formed in writing clear, coherent, and accuruate information is of the utmost importance in almost any endeavor.) She went on to say: "By showing their work, students prove to the teacher that they truly do know how to solve the problem. In addition, showing your work is to the student's benefit because by writing things out, it minimizes the chances of making a careless mistake. I also believe that it is more fair to give students some credit if they show how to do the problem correctly (even if get the answer wrong because of a little arithmetic mistake) than to give them no credit at all for their efforts."

Along the same lines, Matt wrote "We gave the answer and the method equal value. Some people value method and work more than the answer because it shows the thought process behind the answer. Sometimes people can just guess the numerical answer, and have no idea how to actually do the problem! When you are in school, you are trying to learn how to do problems, not just to get the answer by any means. People who are able to identify what went well (and why), and what they want to improve (and how) achieve a higher performance level." I couldn't have said it better myself! He went on to say "The work done to solve a problem is as important as the solution because by showing your work you communicate the process which you went through. By doing this, you and others can see what you did wrong and what you did correctly. I think that we really do learn from our mistakes, and reading someone else's work, we can learn from their mistakes too."

Of all the projects that we did in my Geometry class, this was the one that the students found most interesting. In their reflections, many of the students said that they really appreciated the fact the teacher cared enough to ask them what they thought, and that their ideas about grading mattered. It was one of my favorite projects too, because I found their comments and their insights fascinating; I really believe that I understood their point of view, and shared their troubles and their triumphs.


"No human investigation can be called real science if it cannot be demonstrated mathematically"

Leonardo da Vinci (1452-1519)

Go To Homepage         Go To Introduction

1) Constructions         2) Clock Problem         3) Test Corrections         4) ASN Explain         5) Thoughts About Slope         6) What is Proof?

7) Similar Triangles         8) Homework Corrections         9) Quads Midpoints         10) Quads Congruence         11) Polygons

12) Polygons Into Circles         13) Area and Perimeter         14) Writing About Grading         15) Locus         16) Extra Credit Projects

17) Homework Reflections         18) Students' Overall Reflections         19) Parents' Evaluate Method         20) In Conclusion