David M. Bressoud August, 2007
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In my thirty-six years of teaching, I’ve learned a few valuable lessons. A recent article in The Chronicle of Higher Education, “You Will Be Tested on This,” reminded me of two of them: What you test is what they will learn, and the best way to help students learn is to test them early and often
I learned the first lesson very painfully in a differential equations class early in my career. This was before the age of easy access to computers, back in the days when this course was built almost entirely from examples with exact solutions. I thought I had done a pretty good job of motivating these examples, describing a variety of situations that could be modeled by the various differential equations we encountered. The emphasis, of course, was how to turn an initial value problem into a solution. That is what the homework had been about, and that is what I tested on the midterms.
But for the final exam, I decided that I wanted to see how well my students had grasped the larger picture. Could they take a situation similar to those I’d talked about, model it with an appropriate differential equation, and then solve that equation to gain insight into the original situation? Without much trouble, I found a several good problems that tested their understanding at this deeper level.
The final exam was, of course, a disaster. I had never required my students to construct differential equations from a description of a situation. When I had modeled this for them in the classroom, they had taken it as an interesting aside, but not really what they were there to learn. I had never tested them on it before. They not only did very poorly on my exam, they were angry that I had dared in the final exam to change the contract of what they were supposed to learn in this course.
In succeeding years, I tried setting out my expectations at the beginning of the class: This is what I expect you to be able to do by the time you finish my class. I learned that most students paid exactly as much attention to these course descriptions as I did. If there was something on my list that I never quite got around to covering, no one complained. If there was something else that I inserted, and made it clear to them during the course of the semester that it was important, they accepted it as important. My actions spoke much louder than the words I had used on the first day. In fact, as I came to realize, the whole exercise of laying out course expectations was really for my sake, not for theirs. It served to remind me where I had intended to go when I first conceived this course, what I had hoped to accomplish with these students. My students paid attention to my homework assignments, quizzes, and exams. Those told them what they needed to learn, what this course was really about.
It gradually dawned on me that there is a corollary to “What you test is what they learn.” Students arrive in math class with a set of expectations of what will happen. Most expect to be presented with an assortment of procedures for finding answers. Their task is to learn how to choose the right procedure to apply to a given problem and then to execute it correctly. For most of them, this is what mathematics has always been. So when I began to probe their understanding, asking them to explain what they did and why they did it, challenging them to investigate and solve difficult problems that did not fit any of their templates, they were wrong-footed and upset. Even if I did this in their first midterm exam, even if I had warned them that this was what I would expect, they did not really believe me until they saw it on the exam, and they felt cheated. I was changing the rules on them in mid-semester.
And so I learned to introduce assignments, quizzes, and projects, earlier and earlier in the semester. The first ones did not need to carry much weight, but they did need to count so that students would see that this is what I really wanted them to be able to do. I had finally learned my second lesson: “Test early, and test often.”
I particularly like projects, difficult problems that are started in class, completed outside of class, that require some original thinking and a synthesis of seemingly disparate ideas. I find it very instructive to require the students to write up careful and complete solutions, both to help me understand their thinking and to help them learn how to express themselves in the clear, precise language of mathematics. And, so that they pay attention to my comments, I require at least one draft that I critique heavily. Early in the semester, these are always group projects. This facilitates the formation of study groups, and it helps students who have never done this style of mathematics to get support from their peers. By the end of the semester, the projects may be started in groups, but usually are completed individually. But what is especially important is that the first of these projects always gets started late in the first week or early in the second week of class, as soon as my class list has stabilized. Students learn very quickly what I will be expecting.
In addition, I now will never give the first midterm exam later than the fourth week of class, and occasionally I sneak it into the third week. This gives my students some very quick feedback on how they are doing and a very clear signal of what I expect.
When I decided that I wanted my students to read the chapter before each class, I did not just tell them that this was what I expected. I made part of their grade dependent on it. I now have a general policy of requiring each student to submit a short paragraph that I get at least one hour before class  that answers two of the three questions: What was the main point of this reading? What did you find surprising in this reading? What did you find confusing in this reading? Submitting their answers to at least 95% of the readings earns them all possible points for the semester, usually 5% of the total grade. Not much, you might say, but enough that my students take it seriously. I am rewarded with students who have read the chapter before class, who have thought at least a little about what it says, and who have clued me in on where the difficulties lie for them.
The point applies to whatever your goals for a course might be. Once you have clarified to yourself what you want your students to learn in this class, you need to think about how it can be evaluated, how to hold your students accountable for achieving it, and how to communicate this expectation early and forcefully. Of course, you also need to think about what it will take to get your students to succeed, the intermediate steps that will help them build the expertise you want. I have found that knowing how I will assess their achievement also helps me plot how to help them get there.
 David Glenn. You Will Be Tested on This. The Chronicle of Higher Education. June 8, 2007. pp. A15–17. http://chronicle.com/weekly/v53/i40/40a01401.htm
 Especially for this first midterm exam, it is very important to give students a chance to redeem themselves if they have badly misjudged how to prepare. I usually let the students earn back half the points they have lost on a given problem if they give me a written explanation of where they went wrong and how to solve the problem correctly. This forces students to confront their mistaken understandings and learn something from them. Letting students earn back points they have lost also enables me to avoid falling back on the grading curve, one of the most pernicious practices in education.
 Initially, I had the students send these answers via email. I now use the online text assignment in Moodle, an open source web-based course management system, which makes the entire process easy for them and painless for me. They click on the appropriate icon, a window appears in which they write their answers, and they click to submit. I see a list with time stamps of students who have submitted answers, I click on the student name to see the response, click on the appropriate grade which is automatically recorded and available on an Excel spreadsheet (I grade all or nothing), submit, and go to the next name.
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|David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at firstname.lastname@example.org.|