# What are effective alternatives to a written math exam to evaluate knowledge?

Some students do not do well under pressure or testing situations and cannot demonstrate their knowledge of a math subject with a written exam. On the other hand, I think that students that do not have a good enough grasp of the material are able to do well on a written exam also.

What are testing alternatives that allow an instructor to evaluate a student's knowledge?

For example, maybe having students present problems on the board, as a sort of oral exam, but not calling it an exam so it does not induce anxiety, along with a written exam. Or no exam at all, where the grade is based entirely on homework scores.

Here is a small list of alternatives to an exam. Lets assume you are not constrainted by your university to some specific kind of progress. First lets collect some criteria we want:

• Everyone understood the topic should pass the course.
• Good motivated should should pass with a good grade.
• Students who, e.g., only learned some definition by heart, should fail.
• It should be fair in
• At least as possible anxiety should be introduced (that's in fact the question here. Due to my experience, you would only move anxiety to a different point as long as there is a progress with a grade at the end. If you don't call it exam, the students will know it is some kind of anyway.).

Oral exam

Pros:

• If a question is formulated in a confusing way, there is a chance to ask.
• You can see if someone only learned things by heart or is really thinking about them.
• You can switch to a different topic when there is the impression that the students is really familiar with one topic.

Cons:

• Not possible for big courses.
• If someone fails, you have to say it into the student's face and you have (exept from maybe one witness) no proof of the bad performance of the student (in a written exam, you can let colleagues look at it). Let's face it: Poor students will normally pass the oral exam.
• The impression depends highly on your actual mood and the order of students. At the end you will maybe grade different than in the beginning.
• You do only grade good ideas. If someone is not able to write down anything, this is not even noticed.
• The questions either differ in difficulty or someone can be prepared for your questions.

Pros:

• It does not depend on a one-day performance.
• You can be sure that every topic was somehow included in the grade.
• Everyone has the same questions.

Cons:

• You cannot ensure that the students have done their homework by their own.
• The grade may depend on the TA, compare this question: How to standardize grading across several sections of a course?.
• Students will bargain for points during the whole course.
• Homework should be there for learning and making mistakes. It is okay (and I also recommend this) to have a threshold being achieved to ensure basic knowledge.

Mixture of written exam, oral exam and homework

Pro: You get a good mixture of the pros above and the Cons above will be weaker since the are mixed.

Con: A lot more work for you and your group.

No grades at all, but a written report

Here, you write down some thoughts about the students relying on your personal view (from the lecture, your office hour), the view of the TA, as well as the homework. This report can be shown to other professors in order to apply for, e.g., a master's thesis. Your university will problably not like this since are should give grades in each course. But interestingly enough: Such a model is used in first grades of elementary school and when someone is applying for (tenure) academic position. But normally not in between :)

Visiting the group for (half) a day

In a very very small course, it could be a possibility that the students have to visit to group for (half) a day and talk with everyone from the group. After every student was there, the group meeets and decides about grades.

I have never heard of anyone doing this (it takes a lot of time and distracts everyone from working), but a similar method is used when several people are applying for grants and some comitee visits them for a day or more and decides weeks later (This is the procedure at least in Germany).

• I'm a little confused by this: "Let's face it: Poor students will normally pass the oral exam." I don't see why this is necessarily the case. Can you elaborate as to what you mean? – WetlabStudent May 6 '14 at 5:57
• @MHH: Most of the oral exams start as follows: The student is asked to reflect definitions, statement of theorems and example cases of definitions. Of course, there is asked more, but a big amount are questions which can be learned by heart. Plus, many of the questions are already known due to other students who had the oral exam before. Even if the student knows nothing aside of the definitions and statement of theorems, it's hard to argue with the student and let him fail. This question is related: matheducators.stackexchange.com/q/1345/114 – Markus Klein May 6 '14 at 6:06

In teacher education I had some positive experiences with term papers. Students had to write 10-20 pages where they discussed a topic related to the course and their future teaching. Topics were like "Is there a usefull value that could be given to $0^0$?" and "On the different use and notation of sequences at school and university".

I think this works only, when you want your students to exemplarily go deeper into one question. If the whole course is to be covered, this won't help.

• How do you get to 10 pages on 0^0? – Chris Cunningham Apr 9 '14 at 15:13
• Yes I do. First, discuss why we set $a^0=1$ for $a>0$ (maybe restricted to the naturals). Algebraic arguments (like $a^0\cdot a^n = a^{0+n}$ or the binomial theorem) would also indicate $0^0=1$, whereas argumentation using continuity is ambiguous. My student prooved that $\lim {v_n}^{u_n}=a$ is solvable for any positive $a$. So pro's and con's for $0^0=1$, $0^0=0$ and $0^0$ being undefined are discussed. At any stage, refer to textbooks from school and how they deal with this. In school, reasoning is often built on analogies (like $4^0=1, 3^0=1, 2^0=1, 1^0=1, 0^0=?$). – Anschewski Apr 9 '14 at 15:42

I like to structure all exams with two equally-weighted components: a traditional in-class portion and a take-home portion. The in-class portion generally consists of routine exercises, not too many, and are meant to demonstrate basic proficiency; the take-home portion will typically have five or six multi-part, more challenging questions, from which students are asked to choose a subset (typically $n-2$). I usually give a week on the take-home problems.

Of course the big risk is that students will cheat on the take-home portion. I try to forestall that up front by being very explicit about what is and is not okay: looking up information and ideas in published sources (whether in print or online) is okay, asking someone for help is not (whether in person or online). I have them sign an honor statement affirming that they have understood and complied with those expectations, and I make a big speech about how I am willing to take the risk because I believe they are deserving of my trust, etc. So far it has worked out okay.

Random list of ideas:

• board races - this is not great for grades but is a good evaluation of where the students stand against each other to monitor progress on the topics

• in class worksheets...worth more than a homework but less than a test or quiz

• making study guides - have students create their own study guides in class with example problems that they create and step-by-step solutions, vocabulary, symbols, etc. turn this in for a grade, much like a term paper would be, but more fun. Possibly allow them to work in small groups?

• make your own test - set simple parameters such as you must cover a problem from these five topics we learned this unit and have students create their own tests and answer keys. Let them know it will be graded by correctness as well as complexity.