Inspired by Andy Rundquist, I did oral exams for the first time in my mathematics course for elementary education majors.

The format was a 20 minute one-on-one private interview in my office. Students signed up for appointments by using the Google Calendar Appointments feature. Students were told the questions ahead of time (“explain why you know that the standard algorithm for addition/subtraction/multiplication/division of integers is guaranteed to give you the correct answer to an addition/subtraction/multiplication/division question;” I did not ask the addition question). I gave students a worked-out multiplication problem (using the standard algorithm), but did not work out the solution to the subtraction and division problem.

Here are four things I learned:

- One-on-one oral exams take a lot of time. I have 32 students, which means that I had almost 11 hours of meetings over the course of three days. I cancelled two days of class to offset this, but it consumed my week (which was only a three-day week due to Easter).
- Google’s appoints feature can be buggy. A significant number of students viewed my appointment slot times as shifted by 12 hours, so some students signed up for meetings at 1 am. I am not sure why this happened, although students who used Firefox did not seem to have this problem (Safari and Explorer caused problems for some students, though).
- I learned of some incorrect reasoning on some of the algorithms that I would not have learned if I had just seen written work. In a problem like 203-87, there is a complex “borrowing” that needs to happen. Some students had written work that made sense, but they said (and meant) that you can borrow from the “0” (it just turns into a “9” after you borrow).
- I learned that my students have a really pretty good grasp of the details of why these algorithms work, but they cannot put the pieces together. I think that I have not done a good job of conveying the purpose of asking questions like “why does long division work?” I think that some students do not understand that the long division algorithm can be verified; they essentially still believe that division is defined to be “the long division algorithm” (in spite of the fact that they will give you a correct definition of division if you ask them).

In short, I found them extremely valuable (although tiring). Most students seemed to prefer them to written assessments, too (at least, the ones who spoke up when I asked the entire class about it). It definitely has influence how I will be teaching the next quarter of the semester: I will be focusing on the big picture.

April 13, 2012 at 2:57 pm |

I had my first oral exam this week in a soil mechanics course I’m currently taking. I think I learned more from this 20 minute exam than any assessment prior. I learned that I can be a little lazy in my rigor when speaking opposed to my formal written presentation. Keeping up the rigor when speaking forces a more nuanced look at the subject. There is a difference in thought process when speaking as opposed to writing. When I was forced to verbally discuss my understanding of the material it shined a light on some areas which were not as fully developed as I previously thought. I think the oral exams were as helpful for the students as they were for the professor.

It also reinforced that thinking, speaking, and writing on a board at the same time is very difficult, engendering a little more compassion for lecturers.

April 13, 2012 at 3:02 pm |

Hi Bryan,

That was a nice bit of timing on our part! Thanks for the feedback—you are helping me convince myself that it is worth taking the time to do this a couple times per semester (at least in some classes).

And isn’t it nice when you

learnsomething from an assessment? It seems like your soil mechanics professor set it up well.Bret

April 19, 2012 at 7:28 pm

Assessments, by design, are not teaching tools but I often find I learn a great deal from them. Every time I get something wrong I remember it. That may have more to do with my personality than the process, as misunderstanding a subject I am interested in is very irritating to me. It’s like a bad beat in poker (http://en.wikipedia.org/wiki/Bad_beat). You don’t remember the hands you win, but you certainly remember the hands you thought you’d win but ultimately lose. Now I’m off to get my score.

April 19, 2012 at 7:31 pm

Hi Bryan,

I hope you don’t end up with a bad beat! Bret

April 13, 2012 at 4:27 pm |

This is enlightening, I think, Bret, for all educators. I’m not a mathematician, but I have studied some philosophy of mathematics, a study one might think would predispose me to thinking about the “big pictures” of math. In large, I think this is true. But I was shocked recently (and a little embarrassed) to learn while teaching math to 3rd graders that I was not even aware that division problems may be summarized in the following way: Another way to say “what is 2/3 divided by 4/5?” is to ask how many 4/5 there are in 2/3.

I had only ever thought of division of fractions by the mechanical operation of multiplying by the reciprocal. (In retrospect, that this was a new idea is especially surprising given that this way of paraphrasing division of natural numbers seems intuitive: “21 divided by 7” is easily translated as “How many 7s are there in 21?”) Perhaps the simple complication of fractions muddied my “big picture” thinking.

But the point is that the new way of looking at the problem seemed to present even simple math in new ways. Once I got over the shock, I felt enlightened. I learned something new. I think this only happened by engaging with the symbols in a new, verbal way.

April 13, 2012 at 5:40 pm |

Hi Mark,

You are hitting on what I think is the biggest barrier to mathematical literacy in the U.S.: we have favored mechanical computation over simple understanding for too long. This may have been necessary back when we needed relatively few people to perform difficult calculuations by hand (there were no computers once), but now we want a lot of people to have good conceptual understanding.

I am going to try to blow your mind more: did you know that there are

twotypes of division? I didn’t until the last year of graduate school (in mathematics). One definition is essentially “how many 7’s are there in 21,” but I will rephrase it as so: “If I have 21 cookies and I want to make groups of 7, how many groups can I make?”A second division that has the same answer but is conceptually different is: “If I want to equally share 21 cookies among 7 people, how many cookies will each person get?” Imagine you are an 8-year old trying to solve that work problem knowing only the “how many 7’s are there in 21” notion of division (“how many ‘7 people’s’ are there in 21 cookies?”). The teacher is likely to get frustrated that the 8-year old cannot solve it, but it is really a completely different problem (conceptually).

It is really sad: mathematics is the only area of knowledge where

everythingwhere everything is in principle completely understandable (if humans cannot understand it, it is not mathematics. I am being only slightly hyperbolic here), but we teach it in a way that does not respect this. In fact, I think that we teach it in such a way that students naturally learn to NOT use common sense when doing mathematics.I should stop ranting now. Thanks for the comment!

Bret

April 13, 2012 at 6:16 pm |

You’re right: you blew my mind. I had to read it over 3 times to believe it was a conceptually distinct division! (Or, to put it another way, I read it 21 times and interpreted it equally seven different ways — no interpretation getting any more readings than any other. — JK)

But I’ll bet I had a hard time believing it because the “answer” was the same either way: 3. (I’ll bet I’ve beaten it into my mind that the only thing that matters is “the answer”.)

April 13, 2012 at 7:14 pm |

Hi Mark,

It blew my mind, too, and it took me probably 3 years of teaching it before I became anywhere near fluent in distinguishing them (I still need to pause, so it is not fully automatic).

You also asked a great question: if there are two different types of division, why is there only one way to divide. That is, why does the long division algorithm (or “invert and multiply”) give correct answers for BOTH types of division. Bret

April 17, 2012 at 5:45 pm |

Hi Bret. I just started using oral exams this term and also found google appointments to be a bit buggy. In a class of 12, I found them to be much more efficient (from a time standpoint) than a written exam, but with more than 30 students I expect that I would lose that sense of time efficiency.

April 17, 2012 at 5:51 pm |

Hi Joss,

First, thanks for confirming that Google Appointments can be buggy. I was concerned it was just me.

I hadn’t thought about oral exams being _time_ efficient, although that makes sense. I have a decent number of classes will 12ish students, so maybe I will try that (I might ask my complex analysis class if they would prefer oral exams for their finals). Bret

April 17, 2012 at 6:00 pm

In my case the students were each working on different experiments in the lab so generating 6 different exams and marking 12 was much less efficient than sitting down for 10-15 minutes per group to generate the questions. The students and I worked out their marks together before they left my office so there was no “marking” needed and the feedback was immediate. But I did have one day that it felt like I was running oral exams from dusk to dawn.

April 17, 2012 at 7:51 pm

This seems reasonable and good. I like that the students helped to mark the exam. Bret

June 8, 2012 at 4:41 pm |

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