Sunday, June 7, 2015

Are Exams Useful?

Solo exam
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?

Monday, June 1, 2015

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 43rd 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.

Friday, April 3, 2015

Coding & Probabiltiy


I wanted to spruce things up a little in my grade 11 college math class. My students were working on probability and I was looking to make it more interesting. 

I decided I would have them code games that use probability in Scratch. I choose Scratch because it's easy to use and you don't have to spend a lot of time on syntax. It's also free and web-based, which means it will work on any device that supports Flash. By creating a free account students are able to save their work and they can publish their finished products so that other people can play them.

I started walking students through how to simulate tossing a coin. If you're looking for a tutorial, check out the one made by @brianaspinall here. We spent the better part of a period working on this. The next day I was away, but I left this handout. Students were to 'play' with their coin flipper and make observations about theoretical and experimental probabilities. Once they were finished they had to create a similar program that involved the rolling of a die. They repeated the experiments and then moved on to two dice. The last part of the assignment was for them to create a game using the dice. I was hoping that they would create three games: one that was fair, one that was in the computer's favour and one that was in the user's favour. Upon my return I realized that this was going to take to long. I think next time I will have them choose which type of game to make and to explain what makes it advantageous (or not).

What I liked about the assignment is that students seemed to enjoy themselves. They could be creative. Many of them made some very nice looking dice and backgrounds. They had to do some problem solving when the program didn't work. Best of all, they had a chance to make something with what they learned in math. I would certainly do it again in the future.

Thursday, February 19, 2015

Stacking the Odds in My Favour


I'm currently teaching a grade 10 applied math class. I'm following @MaryBourassa's lead of spiralling the curriculum (Thanks Mary!) and I'm really enjoying it.

Today we had a quiz and a number of my students were quite nervous. Just before class started one girl said to me that she was going to fail the quiz. I reassured her that she was not going to fail. I told her that if she thought about the questions and wrote something down she would do just fine. She didn't buy it. Her response was "No I'm going to fail. I bet I fail. I'll bet you $5 I fail". I smiled at her and told her that I couldn't make that bet because she could easily make things go her way. I was glad to hear her response of "I would never fail on purpose".

As the class worked on their warm-up I thought about how I could guarantee a win on this bet I wasn't going to make. I didn't want to win to get $5 or to say that I was right. I wanted to win to help this student and others who were feeling the same way boost their self confidence. This course is typically comprised of students who don't feel comfortable in math and who don't think they can do math. Today, boosting their self-confidence was my number one priority. As I handed out the quiz I informed them that they would be allowed to use their notebooks. This made many of them feel more at ease and as it turns out, few of them used their notes. I will still get some good information on what they do and do not understand and I will have an opportunity to assess them again at a later date. I'm even toying with the idea of not including a mark on the quiz. I may just provide feedback.

Saturday, February 7, 2015

Sine Law

Last semester I taught the Grade 11 College level math class. I was very disappointed to see that 12 out of 26 of my students had failed the course. Luckily, I get to teach the class again this semester. This means I can make some changes in the hopes of improving my students' understanding. THere is a summary of my first change.

Friday I was teaching the Sine Law. I have in the past created a dynamic geometry sketch. I manipulated it and as a class we noticed how the ratio of the side length to the sine of the corresponding angle was equal for all pairs of angles and corresponding side lengths. For whatever reason, last semester I didn't even show the sketch.

This semester I decided to have the students do the investigating on their own to see what they could come up with. I provided them with a link to the simple worksheet below.



I wanted them to look at the ratios (mentioned above) from a number of different triangles so I had them complete this handout. We completed the first two entries in the table together before I turned them loose. I figured that the table would be fairly straight forward, but I was pretty confident that questions 5 and 6 (where students had to apply what they learned) would be a challenge. Sure enough I had a number of students call me over and say "I don't know what to do here". My response was to have them tell me what they discovered earlier and then tell them to use that information with what was given in the question to set up an equation. That was enough for a number of them to make the connection and do the problems...without any direct teaching. They figured it out on their own. I was blown away.

We had some guests in our class that day. During the activity, one of the guests said to me that this isn't an activity that would be typical in this type of class. I think he is probably right and I think that is part of the reason why the course has such a high failure rate. I challenged my students to learn something on their own and they did it. I think (at least I'm hopeful) that we have established the expectation that students will be active participants in their learning. Now I just need to find a way to maintain that expectation for the remainder of the semester.

The only disadvantage I saw to Friday's class was that many of my students were away. I will summarize the work we did on Friday and give everyone an opportunity to practice. We'll see how it goes.

Monday, January 19, 2015

EQAO Reflection

Our grade nine students wrote their Grade 9 Assessment of Mathematics (EQAO) last week. Often during this time I reflect on the process, because really what else are you going to do for two hours while supervising. This year my thinking wasn't about the pros or cons about the test but rather the way we evaluate it. The test is sent off to be marked provincially but before that happens schools have the option to evaluate it in order to include some or all of the mark as part of the student's final grade. The thinking here is that if it counts for something then perhaps students will take it seriously. At my school we count the test for 10% of a student's final grade. Then about a week later they will write the final exam that counts for 20% of their grade.

The test consists of two booklets that each must be completed in an hour. Each booklet is made up of 7 multiple choice questions, followed by four longer 'open response' questions then finishes with 7 more multiple choice questions. Once the second booklet is completed students are asked to complete a questionnaire.

My observation has been that more often than not students come into the test under prepared and it serves as a bit of a wake up call to them. They then (hopefully) use the remaining classes to prepare for the final exam.

This year I have decided that I am not happy with counting the test for any portion of the students' final marks. In fact, my students did so poorly that after the fact I told them that I was not going to count it at all towards their mark and here's why:


1. Time

Many students did not have time to complete the test. They had an hour to complete each of the two booklets. For the second year in a row my strongest students did not complete the booklets on the first day. These students were very concerned about the impact it was going to have on their overall grade. Rather than providing incentive to do well it caused a great deal of anxiety. As a math teacher my goal is to help students reduce their anxiety towards math not contribute to it. I also try to evaluate what a student knows and does not know. If a question is left blank I have no idea if it was because the student ran out of time or because they did not know how to do it. By removing time from the equation I can make a better judgement of what the student know.


2. Multiple Choice

I have decided that I disagree with the multiple choice questions. They obstruct my view of what the student does or does not know. Some students will get the correct answer by guessing. Others will get the incorrect answer by guessing. In either case, I am unable to see the process that allowed them to arrive at their answer and as a result I am unable make a true judgement of their understanding of the material.


3. Feedback

I don't know much about the official feedback students get so if I'm wrong here let me know. I believe that tests get marked in the summer (the rest of the cohort will write in June) and a mark is returned to the students in the fall. This is far from immediate feedback and is anything but descriptive. Not very useful in my mind. As a teacher I can mark the work, but I'm not allowed to copy anything. This means that I can't show students where they went wrong. I can tell them that they messed up on the bicycle question but unless they can see where, I'm not sure that's useful.


4. Justification

I'd be hard pressed to justify any mark to a student or a parent given that the tests get sent off, never to be seen again. Students should be able to look at their marked work and question my judgement, which is sometimes right and sometimes wrong. In fact, I enjoy when students start questioning my evaluation as it often brings out what they truly meant to write or allows me to better understand their misconceptions.


5. Rationale

When students ask why the test has to count for a portion of their grade I struggle to give a valid reason. I typically say something along the lines of "If you're going to spend two days writing it, we may as well give you some credit for it". It's not an answer I'm comfortable with but it's all I have. One of the reasons I'm not comfortable with it is that the vast majority of my students perform much worse on the test than they do on the final exam. We could probably discuss what that says about my teaching, but let's save that for another post. The real reason that we count the test as a portion of a student's grade is that we believe that this will make them take it more seriously, which means they will perform better, which will make the school look better. Given that twelve out of eighteen students in my colleague's class said on the survey at the end  that they didn't know if the test was going to count (and yes he did let them know on numerous occasions), I'm not sure that counting it is a good motivator. Besides, is this in the best interest of the student or the school?

I'm curious to know whether counting the test as a portion of a student's grade makes them perform better. What does the data say? Do schools that count the test outperform those that don't? Is this information publicly available? Are there any schools that don't count the test?  Or does everyone count the test so that they don't look bad? Is this in the best interest of the students?

What does your school do about EQAO testing in grade nine math?

Tuesday, November 18, 2014

Triangles in Scratch

The video below is a follow up to the one I made here. This one is not that much different. Instead of drawing a square I show how to draw an equilateral triangle. To me the interesting part of the tutorial is having students play around in scratch to see if they can create an isosceles or scalene triangle. I haven't had a chance to do this in a class yet, but I'm hoping that there will be a lot of trial and error and eventually some students will get close enough. Close enough that they will be able to answer the questions "What is the sum of the angles in any triangle?".

The second challenge is to have students draw a regular pentagon, hexagon, etc. In order to do this they will have to determine the sum of the angles in each of these figures. We've done enough visual patterns that I hope this will be easy for them. I'm hoping this will be a fun way to cover some of the geometry in the grade nine math courses.

Give it a try.