Mathematical Uncertainty in Classroom Assessments

In simple math, an answer is either right or it is wrong, right? If the problem is 2+3, then the answer is "5", end of question. If the student answers "6", then you probe a little and find out what the student was thinking. In either case, the question is a straight-forward measurement of the student's ability to count.

Let's say the student moves to simple Algebra and is asked to solve the problem, 2+x=5. The traditional answer, of course, is "3" assuming "x" is a positive integer. Yet just stating the answer, that is grading the assessment without review and reflection, may not correctly evaluate a student's ability to handle the material. The student may have remembered the previous questions, may have had a lucky guess or may be able to handle positive numbers and be blown away by negative integers. Can you tell if a student understands Algebra from the answers to one question? Probably not.

So, we move to multiple questions with multiple answers. I know this drives students crazy (especially the ones who just want me to tell them what is the right or wrong answer), but I don't care about the answer as much as I do for the process and discovery leading up to the answer. That's where learning takes place. If we want facts regurgitated, we google them. We don't need students to sub as smart phones.

A common gripe I hear from GED students is that the questions on the credentialed test are not the same as those in class or on the practice tests -- well, yeahhh. Novice students expect a 1:1 alignment of study questions to test questions, which is not going to happen. There's an Urban Legend in teaching circles that after a GED math test a student complained bitterly to a teacher that the class did not teach the algebra material that was on the GED test.

"What do you mean?" the teacher asked.

"In class we learned that if x+2=5, then x=3. And if x-2=1, then x=3," the student said.

"Yeah?" said the the teacher, not understanding the crisis.

"But, the test asked what was the answer to a+2=5 and n-2=1," said the student. "You didn't teach us about 'a' and 'n'. You only taught us about 'x'."

As a colleague once told me, the classroom curriculum is the map and the assessments are the compass. Sure, we teach the facts, the topography of the course, the scale of the region and annual migration routes of indigenous inhabitants, but more important we teach how to use the compass when you find yourself in unfamiliar territory. Assessments are as much a learning tool as are Power Points and study time.

Problems on the GED test, in college classrooms and in job situations are rarely as simple as 2+3=5 whether the variable "x", "a" or "n".  More often a problem is x+y=z, where 0<x<100 and y is a squared multiple of z depending the latitude of the inverse proposition.  Not only don't you know what the initial quantities are, you may not even know what the answer is supposed to be. Keying in on answers alone assumes that right answers or wrong answers are the most important thing in learning. They aren't. What's most important is the process students use to divine the answer. The best assessments that discover that process are as creative, varied and reflective as is the problem. The answer key to those assessments is rarely in the back of the book.