Points on a test are not gold coins!

anxiousOne problem that Occupy Math has in teaching his first year courses is that many of the students have been trained, by their high school experience, to believe that a math class is a game that is scored in points with the goal of a grade. If it weren’t actually true in some of their high school classes, it would be nonsensical, and the whole notion is counter-productive to the goal of getting an education. Every year, Occupy Math has some students who are trying to do just enough work to pass the course, no more. Many of them flunk because a math course builds technique upon technique. What appears to be the correct level of effort for a D near the beginning of the term is actually preparation for an F or an F–. These grade-management tools from high school are also used as fear-management tools; instead of engaging with math, the student tries to scam a passing grade and so avoids the math. This has all sorts of bad downstream effects. In this week’s Occupy Math, we want to look at the issues of effort, fear, and effective teaching and learning.

The most ironic thing about trying to work just hard enough to get a D is that, in addition to the risk of gauging the effort incorrectly and flunking, keeping up and trying to do all the work is usually less effort in university than a well-crafted D-strategy would be. Besides, a full-participation strategy often yields an A or a B. It’s important that students realize that the sense they developed of how much work is needed to get a given grade in high school will be misleading when the student makes it to university, where they will face an entirely different scale of effort and reward. In addition, students who employed the minimum-effort technique in high school often have deficient preparation. Bad news all around.

Instead, just learn the material.

A retired colleague of mine, Jack Weiner, taught our first year courses at the University of Guelph for decades. Jack used an on-line quiz system that could generate quizzes on the fly. Since we were using on-line testing, we needed a different quiz for each student. Jack did something pretty clever with this: he let students take the quiz over and over and keep the best grade. Sound like a give-away? Think Jack is an old softy? One of the students put it this way: “I thought you were giving away grades, but afterwards I realized you were tricking me into practicing.” Occupy Math has tricked students into working hard as well, back when his classes were in a room instead of an auditorium.

The effect of doing all the work (and then some) is that you learn the math and then the class isn’t all that hard. Occupy Math is sure of this, but with one important exception. When fear becomes a part of the situation, you can get students that honestly believe they’ve worked on an assignment for hours and made no progress. The problem is that you can’t work for hours on the kind of assignments that are handed out in a first-year math class and not make progress. What is actually happening is that the student is stewing in fear, possibly expressing itself as exhaustion or apathy, and not working on the assignment for hours. This leads very naturally to the question, what can you do?

Dealing with fear, from both sides

Lets begin with the usual advice before we get to what the instructor can do.

  • Work with other students, form a study group.
  • Go to office hours. How well this works depends on the instructor, but it often helps and a bad experience with one instructor gives you no information about what might happen with another instructor.
  • If there is a help center, consulting room, or learning center, go to it and get help when you’re stuck. Occupy Math’s institution has a large room with mathematics and statistics graduate students in it, open 6-7 hours a day. These are people whose only job is to answer your questions.
  • Ask your guidance councilor what resources there are to get help. It’s their job to know and tell you and they would much rather tell you where the help is than make up a plan to get you off of academic probation.
  • Take care of yourself! Enough sleep, remembering to eat healthy food, and responsible drinking are all part of this. Work-life balance is important and being in good enough shape to do your job is part of that.

Let’s turn to techniques for teachers. Occupy Math took a two semester course in the mathematical foundations of statistics when he was an undergraduate at the University of Kansas. The instructor had an interesting way of giving tests. He handed out ten questions ahead of time. You could prepare any way you wanted to, and, when test time rolled around, he let you exclude some of the questions. He would then pick three of the remaining questions, but there was a catch. Your score on the test was multiplied by the fraction of questions you allowed. If you exclude two questions, then you’re only allowing 80% of the questions through. In that case, a perfect paper gets 80%. Since it was a statistics class, the problem exclusion mechanism was part of the test. Someone that understood statistics would always let all the questions through because that maximized the average outcome. It also helped that the instructor wasn’t a bastard.

Even though the correct solution is “let all the problems through”, a risk-adverse student might exclude a problem that they have no clue on. Leaving aside how it is possible to be clueless on a problem you were given a week ago, this speaks to an important point. When Occupy Math gives an old-style exam, one where there are questions and you have to answer them all, it is common for a student to object “you asked the one question I couldn’t answer!” There is an incredibly cruel way to deal with this: offer an alternate oral exam where the instructor asks and the student answers. It is seldom the case that there is only one question a student cannot answer.

While giving an alternate oral exam might work as a method of stifling complaint, and reducing obnoxious students to quivering masses of Jello, it’s not really an effective teaching strategy because it increases fear. An instructor might, instead, structure an exam in the following fashion. Tell the students to “Answer four of the following seven questions.” This is a cruelty-free way of making it impossible to claim the test contained the one question a student couldn’t answer. It also means that the students will choose problems they can do which pays a huge bonus when it is time to grade the test. Writing such a test is more work — you have to chose seven questions so that four of them cover most of the topics from class — but it is possible. The net total work is usually better because of the easier grading.

Fear from the Instructor’s Side

Jack Weiner’s technique of having the student take the quiz over and over until they get it, Occupy Math’s game “Sports Day” in one of the links above, and the technique of permitting students to select four of seven questions have two things in common. The first is that they all reduce fear. The second is that other instructors complain that these are techniques used by people that are soft, that these are giveaways. Let’s stop a minute and look at what we are giving away: points. There is an embedded assumption there that points are like gold coins — precious and to be given only upon unequivocal demonstration of merit. Occupy Math thinks that assumption is, itself, counter-productive. It’s directly in line with trying to do just enough work to get a D. The fear of giving away points is a disease of instructors.

Here’s the scoop. Points are not real, they are informational constructs that should be used as effectively as possible to get the students to learn. If your system gives away some points, but gets a good read on how much the students have learned, that is so much better than creating a high-stakes game with points in the role of poker chips. Where possible, attention should be focused on the content of the class, not the structure of the rewards system. Points, unlike energy, are not conserved. A teacher who is seen as generous and helpful extracts better performance than one who is seen as a distant authority figure to be tricked, deceived, and cheated. You do not want to be “the man”.

Occupy Math will conclude with a technique he uses to keep attention focused on course content. On the first day of class I warn my students that the question “Will that be on the test?” is always answered in the affirmative, but that Occupy Math reserves the right to lie. In spite of this, students will ask the question. It is important to say “yes” with a small pause before the word, an ironic look on the face, and a special voice that suggests the question is slightly improper. What will be on the test should, of course, be listed explicitly in the course syllabus.

Occupy Math hopes this discussion of fear and techniques for combatting it has been helpful or, at least, thought-provoking. Do you have fear-reducing teaching techniques that you would like to share? Occupy Math loves being able to do posts that are based on reader suggestions. Please comment or tweet.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


One thought on “Points on a test are not gold coins!

  1. I always enjoy your posts, but this one in particular really struck a chord with me. I am a HS math teacher and I teach the “lowest” level of 9th grade math (and 11th grade math). The majority of the students think that they are “not a math person” or that they are/never will be “good” at math. I am always trying to think of ways to get them to remove their fear from their attitude to the class.

    One of the strategies that I’ve started employing this year is to tell them ahead of time the one open-ended question that will be on the test. I give them this question as part of their review packet and I also give them a sample answer to use as a reference. (For example, for their recent test, I told them that they would be asked to think of their own linear equation, f(x), and their own exponential equation, g(x), and to answer when f(x) = g(x), f(x) > g(x), etc.) It does make it a little harder to grade (because everybody’s answer is different), but it gives them at least one question that they are (hopefully) prepared for, and that involves skills that should be helpful to them in the rest of the test.


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