Discrete Math - Day 39

Today the class had its mid-term exam. The structure of the exam is based on brain research. Specifically, I drew ideas from How the Brain Learns Mathematics, by David A. Sousa.

The mid-term exam is broken into four categories: number theory and cryptography, counting problems, polygonal numbers and finite differences, and probability. Within each category there are four problems. Two of the problems are at a level that I would expect every student to be able to complete with a certain level of success. One problem is at a level that I would expect a moderately successful student to be able to complete. The fourth problem is at a level that I would expect a successful student to be able to complete.

In total there are eight problems at the first level, four problems at the second level, and four problems at the third level. To obtain a C-level grade on the exam, a student must complete six of these problems with a grade of partially correct (P) or essentially correct (E). To obtain a B-level grade, in addition to successfully completing six first level problems, a student must score P or E on two of the second level questions. To obtain an A-level grade, the student must meet the requirements for a B-level grade and complete on of the third level problems with a score of P or E. In addition, for a B or A, at least half of the problems must be completed with a score of E. To ensure that students do not avoid a particular class of problems, they are also required to successfully complete with a score of P or E one problem from each category.

Although this may sound confusing, it provides students choice in the problems they complete while ensuring that students demonstrate a specific level of knowledge and skill regarding the content. I use a color-coded spreadsheet to record problems attempted and problems completed successfully. An E score reflects that the problem is essentially correct, although there may be minor errors that result in an incorrect number solution. A P score reflects that the student was pursuing a productive approach to a problem solution but either was not able to complete their work or left out a component needed to answer the problem fully. An score of incomplete (I) reflects a solution that shows some inkling of what should be done but little indication that the attempt would actually lead to a solution. An I is also used whenever a solution records a numeric solution without explanation or justification. Finally, an X indicates that a student wrote something for an answer that shows no understanding whatsoever of the problem or its solution.

By using this spreadsheet, I can verify that the requirements for a specific grade level have been met. I can also use it to self-reference across students to ensure that scoring is consistent. Finally, I can see a summary of which problems were attempted, how many attempts were successful, and how many attempts were not successful. This allows me to evaluate the quality of the questions and the appropriateness of the questions as to their assignment to a difficulty level.

For example, last year I had a problem designated as first level, meaning I thought that the problem could be completed successfully by every student. As expected, a high percentage of the class attempted the question. The results showed that very few of the class completed the problem successfully. At the same time, a problem that I felt would be very difficult for most students was classified as a third level problem. In this situation, I had many more attempts for this problem almost all of which were successful. The spreadsheet allowed me to re-evaluate the assessment questions and make changes to the test that would better balance student capabilities and problem difficulty.

I do allow students to make use of their notes during the exam. My thinking is two-fold on this matter. First, the questions on the exam are not rehashes of problems worked on in class. The problems, for the most part, are designed to extend student understanding or to have students apply what they have learned in new ways. There is little that a student could pull directly from their notes and record as an answer to the questions on the exam.

The second part of my thinking reflects authenticity. When a professional mathematician is working through a problem, they have materials available to them for reference. It is rare that a mathematician would make a connection to something they learned a while ago and then say to themselves, "Oh well, I don't remember that math, so I guess I'll just skip the problem." No, they will reach over and pull a reference book or go online and skim through material that will help jog their memory of how the math they need works. I believe my students should have the same access as they work through unfamiliar problems.

Some students will complete the minimum number of problems while others will do additional problems to cover themselves. Today was a 90 minute class, and about a third of the class took 80-90 minutes to finish up their exams. Less than five students finished in 45-60 minutes.

Here are some additional observations about this test structure and administration. First, students do use their notes thoughtfully. You will observe students thumbing through their notes and taking time to actually read what they recorded, thinking about the examples and what they did to get to solutions.

Second, students do not complete the test linearly. They jump around. This has two impacts. First, I can leave students sitting in groups because they end up working on different problems and sections. The risk and amount of cheating with this test design is minimized. It is also easy to grade students who sat together to see if they completed all of the same problems with the same answers. My experience shows that cheating has not been an issue.

The fact that students jump around in answering questions and are writing their solutions on blank paper has changed my grading practice. With a traditional test, I would grade one page at a time, going through every student's work. Then I would proceed through the entire class, grading the second page. This would help ensure consistency of my grading. This process is not possible since the tests are not completed linearly. Now, I assess an entire test for one student and then move to the next student. I only record E's, P's, I's, and X's next to each response and don't worry about a numeric score at this point. Once the entire class is assessed, I record the results in my spreadsheet and then assign a numeric score. The spreadsheet becomes my mechanism for consistency.

Third, assessing with E's, P's, I's, and X's necessitates that I have a scoring rubric for each question. What does an E response contain? What mathematics should be demonstrated in the response? How much assistance and guidance would a students need to reach a correct solution? These questions help guide me as to what I should be looking for and considering as I assign my assessment to a problem. This does not mean I have an extensively written rubric for each question, just that I have a solid understanding of what I am looking for in a response to each problem. This helps provide consistency as I work through the exams.

Finally, I have less grumbling about a test being unfair. Students typically respond that the test was difficult or hard but that it was fair. This is important as students have a choice as to what they tackle. Because of this, students realize that their choice of problem may not have turned out to be what they expected. This reflects on their level of understanding. It becomes a self-assessment that maybe they don't know this specific topic as well as they thought. The result is a sense of difficulty versus a sense of fairness.

For anyone interested, I can send you a pdf version of the exam and Microsoft Excel spreadsheet template for recording exam results. Just complete the contact form with your request.

My school is headed off to Spring Break and we won't start back up until April 2nd. I'll pick up with the class then.

Visit the class summary for a student's perspective and to view the lesson slide.