Are Exams Useful?

Solo Exam By: Xavi

Last week our district had Damian Cooper in to talk about assessment. The plan is to have two more follow-up sessions in the fall. I did not attend the session, but have talked to a number of people about it. The one thing I heard over and over again was that there were lots of big ideas but hopefully in future sessions he will provide some more concrete details about implementing his ideas.

I think for those that are interested in what Mr. Cooper said, there's no reason why the conversations about assessment can't start now. I had one such conversation with a colleague the other day. He told me that as he started to put his exams together he wondered "Why?". His point was that if students had already been assessed on certain expectations, what was the point of assessing them again on the same topics. I thought this was a valid point. We discussed it for a short period between classes but I think it's a discussion that could have gone on.

What more can you learn about a student's learning in an hour and a half to two hour exam? One of the options we discussed is the possibility of having targeted exams so that students could show you that they have improved on a topic that they didn't master in the term. This could be as complicated as individualized exams (which would be a lot of work for the teacher) or as simple as giving everyone the same exam and giving each student a list of questions that they must complete on that exam. The goal would be for students to show the teacher that they have gained the understanding that they lacked earlier in the semester. If you decided to go this route you'd have to be a little careful with the weighting. In Ontario 70% of a student's grade is to come from their term work while the remaining 30% comes from a combination of exam and/or summative activity. You wouldn't want 30% of a student's mark to be determined by questions that they struggled with early on. But with a little tweaking I think this could be an effective approach.

Another option we discussed was to make the 'exam' a reflection for the student. It might be a written exam or perhaps in the form of an interview as described here by Alex Overvijk.  A reflection might involve questions such as: What was the most useful topic we covered? What was the most difficult? How do you see using any of the ideas from the course beyond the course? I'm sure there are a lot more, many that would be much better than these but the idea would be to get students thinking and reflecting about their learning.

One final option that we discussed (I'm sure there are many others) was moving away from an exam to a culminating activity that ties together multiple (perhaps all?) strands from a course. This could allow for some creativity and eliminate the time crunch of an exam. It could give the teacher a sense of how much students have grown over the course.

Up until a week ago I thought that exams were crucial but as I've thought and talked about it over the course of the week I think I would be comfortable without one. Here are some of the concerns that I've heard about eliminating exams and my questions about those concerns.

1. Students need to know how to write exams for post-secondary. This may be true but do we need to subject all students to exams. I know that many college courses don't have exams and it appears that some universities are giving far fewer exams. At my school fewer than 20% of students go off to university. Does it make sense to subject every student to learning how to write exams when so few of them actually will? Perhaps exams could be part of the courses for university bound students.
2. Exams provide a check to see if students still know the material or to make sure they really understand it. If a student was able to cram for a test without really knowing what was going on, isn't it possible they could do the same for an exam? How much of the material from an exam is retained by students a month after the fact? 

What are your thoughts about final exams? Are they necessary? Should we be eliminating them? Or should we be looking for a more effective model for exams?

Visual Patterns - Visually

I use Fawn Nguyen's visual patterns a lot in many of my classes. I really enjoy them since the get at a lot of mathematics in different ways. Students really struggle with them early on but get the hang of them before too long.

Today I tried this one.

My grade 10 students quickly realized that the pattern was not linear but a number of them still wanted to represent the pattern with a linear relation. They struggled for a bit and then we talked about how we could use an area model to get something like this.

They realized quickly that the height was just the step number and that the widths were growing linearly. We found an expression for the width in terms of the step number, then multiplied by the height to find the number of helmets. This was a great way to combine linear and quadratic relations.

I had asked students to find the an equation that represents the pattern and to find the number of helmets in the 43^rd step. The best part about this pattern was that before most students had even started working on it, one of my students, K.,  who struggles to write stuff down was madly working away on his calculator. He was clearly working on the specific case rather than the general case. After I had given some time for everyone to work I brought the class together and asked for some ideas. K. was the first to raise his hand. His solution was essentially "multiply 43 by 43 and add 43". He explained why he thought this was the solution but it was clearly over many of my students' heads. I left it out there and took other suggestions. We worked at coming up with the solution algebraically and came up with #helmets = 2n²+n, where n is the step number. As we finished I noticed the similarity between K.'s solution and this one. K.'s solution was # helmets = n²+n. I looked at the image again, knowing the algebraic solution and trying to visualize how K.'s solution fit in. As I looked I saw this.

I was blown away! One of my students was able to see most of this in a matter of seconds. As a math teacher my default tool seems to be algebra, but this visual solutions is much slicker. With the help of my students I think I'm starting to get the hang of doing these visual patterns visually.