By Lew Ludwig

Early in my career, I adopted a test structure worth one hundred points. The test consisted of five-point and ten-point questions to keep the math simple. The twelve or fourteen five-point questions were generally more straightforward, while the three or four ten-pointers helped sort the As from the Bs.

Over time, my A students did indeed correctly complete the ten-point questions. My stronger B students would get partial credit, maybe seven or eight points, but the rest of the class did abysmally – maybe a charitable two or three out of ten points.

Did my other students not grasp eighty percent of these more advanced questions? No, the more advanced questions required an extra layer of understanding – I’ll begrudgingly use the vernacular of “trick” – to proceed with the question. Stymied by this additional layer, my average students could not show their general understanding of the concept, and some even left the question blank.

These ten-point questions did determine the As for the test. But, they discouraged and even intimidated the average student, leading to very few Bs in the test distribution and more grades of Cs or below. I had intermediate students with an average understanding of the material, earning a D or lower due to these high-stakes questions.

Upon reflection, I recalled a technique used (last century) by my math instructor, Maureen Doleman. Instead of having one high-stakes question on a concept, consider having different levels of questions on the same topic. For example, a ten-point question (hard) can have an accompanying seven-point question (more straightforward). Both questions should test the same concept but at different levels. Grade the student who completes the ten-point question out of ten points. However, if students do not make headway on the ten-pointer, they can complete the seven-pointer instead. In this case, a student's maximum score is seven out of ten. My students call these types of questions the seven-ten split.

Before you ask, a student must make a choice and complete only one of these questions. In this case, I will grade the first solution presented. And no, I will not grade each and choose the better score if you do both.

At first glance, a seven out of ten score seems rigid. But considering the alternative, an average student is more likely to earn six or seven points out of ten, instead of the charitable two or three as before. Now my grade distributions on tests better reflect my students' understanding. Moreover, student anxiety about these more challenging questions is significantly reduced with the “safety net” of the seven-ten split. I found students are more willing to struggle with the ten-point question for a while, then cross that out and move to the seven-pointer if stuck.

Upside:

• The distribution of test scores better reflects student understanding.

• Student anxiety decreased.

• Students are more willing to persist in challenging questions.

Downside:

• I now have an additional question to grade.

I did have one “conservative by nature” student who would get a nearly perfect score on the five-point questions but only do the seven-pointers. The student was apprehensive about taking a risk on the ten-pointers. After several encouraging conversations, the student attempted a few 10-pointers by the end of the semester and always earned more than seven points on those questions.

Lew Ludwig is a professor of mathematics and the Director of the Center for Learning and Teaching at Denison University. An active member of the MAA, he recently served on the project team for the MAA Instructional Practices Guide and was the creator and senior editor of the MAA’s former Teaching Tidbits blog.