Monday, November 14, 2016

Exploring Polygon Relationships in Scratch



My grade nine class is currently investigating relationships between two variables. I wanted to find some interesting data to look at but also wanted it to be connected to other parts of the course. It occurred to me one day that the geometry portion of the course has students making connections between the number of sides in polygons and the sum of the interior and exterior angles of polygons. This seemed like a good relationship to look at.

I figured this would be a good opportunity to do some coding. I had students visit the Scratch website. A couple of students had used it before, but for most of them this was a new experience. We talked about how you might give somebody clear instructions to walk in a square then we coded those instructions in Scratch and before long everybody had a square. I then instructed students to create a regular triangle, pentagon, hexagon and heptagon. Upon completing each figure they completed the table on the handout.

After collecting the data they proceeded to investigate the relationship between the number of sides and the sum of the interior/exterior angles.

It was fun to watch them. Some started working by trial and error, others were able to make some connections right away. Many who found a solution quickly helped others make the same connections. This was a great way to integrate coding with the relationships and the geometry section of the course.

This activity had me thinking about how I might use a similar activity to have students draw triangles that were not equilateral. It seems like it could be a good application of the sine & cosine laws.

Here's the handout:


Sunday, October 16, 2016

When The Class Bombs A Test

Photo by Wendy Berry

Last Thursday I gave a test to my grade 12 Advanced Functions class. It was our first test of the year and the results were disastrous. My first hint that things may not go well came the day before the test.

Typically the day before a test students in this class keep me busy for what seems like every minute of the day. Some will come in before school and ask questions. Some will sit in my room during their spares and work so they can ask questions. There's usually a flurry of activity at lunch with students working in groups at their desks or at the board. My prep period becomes an impromptu tutorial for students lucky enough to have a spare at the same time. This year, I had a student or two at lunch time and one who stopped by for a few minutes during my prep period.

While students wrote the test I could tell things weren't going well. Some were taking the long way around for a lot of the questions, which meant they would need extra time (apologies to their period two teachers). Many of them seemed to be struggling. A number of them asked if I could drop this test mark or if we could do a rewrite as they handed the test in. Not a good sign.

I wasn't going to look at the tests that night, but I was curious to see what the results were actually like. My suspicions about the class as a whole doing poorly were confirmed. The marks were terrible. I spent a lot of time thinking about why they were so bad. How much of it was failure on my part? How much did they need to take ownership for? How could we rectify the poor result?

Without having enough time to come up with a solid plan of what to do next I decided that my number one priority had to be for students to master the content. The next day I put them in groups of three and gave each group a copy of the test. I had the groups work through the test at the boards. I was able to circulate, listen to the conversations and provide some leading questions when they were needed. There were some great discussions, problems solving and peer teaching going on. I think most students learned at least a little and some learned a lot.

The logical thing to do seems to be to have a retest. Past experience has shown me that the results from retests are generally not all that much better than the original test. Students have good intentions but then run out of time to prepare so their marks improve very little. I think that walking through the tests in groups helped but they won't be writing the test in groups. To help ensure every student who rewrites the test is well prepared I have decided that I will meet with each of them to go over their test. I will ask questions that will help identify what they know and what they need work on. Hopefully this gives them a list of topics that they should go over in preparation for the test.

It seems like a lot of work but I'm hopeful it will pay off. What strategies do you use when tests or other forms of assessment don't go the way you expected?

Thursday, September 8, 2016

Can We Just Take Notes?

Frustration by Sybren Stüvel
The first week of school is drawing to a close. After seven months off it's good to get back into the swing of things. I was greeted warmly by students (some I have taught, others I have not) and staff upon my return. My goal for the first week was to make it fun, or at least somewhat enjoyable. So many students hate math...or so they say. I wanted to destroy this notion during the first week.  I figure if I can tilt a student's perception of math in the positive direction at the beginning of the semester at least we'd be off on the right foot.

I wanted to minimize the amount of talking I did and maximize the amount of talking my students did. I had them working in groups, often up at the board. I was more interested in working on the mathematical processes and setting the tone for a collaborative environment than covering specific curriculum expectations.

We did some estimating, some visual patterns, some problem solving, some data collection and played a game. It was great for me to be able to spend most of my time circulating and listening to the conversations that were taking place and asking questions. I was enjoying it and it seemed as though most of my students were as well. Some of them would get frustrated at the problems we did. Many were able to overcome that frustration and feel the pride that comes from conquering tough problems.

Today I received mixed, unsolicited feedback from every class about how things were going. How did they know I wanted feedback? A couple of students from my grade nine class and one from my grade eleven class all said something along the lines of "You make math fun. Last year I hated math. Now I like it". On the flip side one of my grade nine students asked if we were going to be taking notes in the class. She looked relieved when I told her that we would eventually. Finally, from a number of grade twelve students today: "Can we just take notes? I don't want to do this group work and problem solving". As it turns out I was going to summarize some of the work we had done with a note towards the end of the class. I was dreading it. It was a boring note as the two people who fell asleep would probably attest to. Why would anyone want to do this rather than being an active participant?

The grade twelve comment is the one that had me thinking the most today. I kept wondering what we have done in our school system to make students want to sit around passively, hopefully, soaking up information. I couldn't help but think that we have trained these students to sit quietly at a desk, listen to a recipe and then follow the recipe a bunch to practice it. They would rather do this than think independently or solve interesting problems. It seems that some of my students don't want to experience productive struggle and the sense of accomplishments that comes with coming out the other side of that struggle.

Don't get me wrong. My students are great and I think this is going to be an excellent semester. I believe that many students are very accustomed to (and good at) 'playing the game'. You know the game I mean: show me what you want me to do, help me figure it out when I get stuck, test me on it and give me a good mark when I give you what you want. They know the game well and many of them are very good at it. When we as teachers change the rules of the game the students who are good at it (often the high achievers) get very nervous. They are still going to do well, but they're not as confident about it.

My grade twelve students are likely the students who will experience the most varied teaching methods when they go off to university next year. They will have lectures, labs, group project (formal and informal), open ended projects, etc. I really feel that they have the most to gain by experiencing different teaching methods and yet they seem to be the most reluctant.

One thing is certain. There will be more problem solving and group work throughout the semester! I can't wait.



Wednesday, September 30, 2015

A Shift in Formative Assessment

Up until this year I have struggled with formative assessment. I get the idea behind it. I know that it's designed to inform both the student and teacher of where students are in their learning. As a teacher I can then adjust my teaching based on what students do and do not understand and hopefully students can focus on what they don't understand. 

In theory it all sounds good but I felt that there was always a huge hurdle to overcome. How could I get students to take formative assessments seriously if it wasn't going to count? We've all heard it before: "Does this count for marks?". To me this question translates into "How much effort do I need to put into this?". I didn't see much point in using valuable class time on an assessment if students we're willing to give it their best shot.

This year I have decided to change my perspective on formative assessment. I realized that formative assessment isn't for me at all. I know how students are doing on certain topics based on my observation and their conversations. I don't need a mark to justify this. The purpose of formative assessment is not to give me feedback, it's to give my students feedback about how they're doing. It should provide them with the tools they need to take the next steps in planning their learning. Following the assessment a student should be able to say "This is what I need to do in order to be successful".

A couple of weeks ago I decided to give my students a formative quiz. I let them know that the quiz didn't count for anything. I also let them know that the purpose of the quiz wasn't to inform my practice, it was to let them know how they were doing. The big difference this time is that I informed my students that there would be no marks on their quizzes. I decided that marks were a distraction from the learning. The purpose of formative assessment had to shift from grades to learning. Instead I provided only feedback. I circled things, underlined stuff, drew arrows and asked questions to help guide their thinking.

Learn by GotCredit
The result is that when the quiz was handed back students didn't just look at their marks and file the quizzes away (either in their binders or the recycling box). There were no marks to look at. If they wanted to see how they did they needed to look at the questions and read the feedback. Reading the feedback led to them asking questions amongst themselves and if needed asking for my assistance. Some of them were upset about making silly mistakes, others were trying to make sense of the topics that they didn't really understand. Not only were they learning, they were learning from each other. This slight shift in focus for formative assessments has made them far more valuable in my class.

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.