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.

Playing With Rectangles

I'm currently teaching my grade 9 students about linear relationships. We create scatter plots, draw lines of best fit, use the information to make predictions and so on. As we came to the end of the unit I felt as though we hadn't done enough. I felt that somehow it would be far more interesting if we could connect this section of the course to another section. The measurement unit seemed like a simple connection.

I gave students 12 straws and asked them to find the rectangle that would give the largest area. They messed around with the straws, made tables and graphs and determined from their graphs what the largest possible area was. Next I gave the 12 linking cubes and asked them to create the rectangle with the smallest perimeter. Again they played, created tables and a graphs but the solution wasn't as obvious.

None of this work is ground breaking or much different from what I have done in the past. The only differences were that I cross pollinated (some might say spiralled) the units. I think this helps show students that mathematics is interconnected, that it's possible for units to have a common thread. The other difference was that I physically gave them objects to manipulate, which is different from how I taught this before. In the past I would have them draw out rectangles. I think something gets lost here. It was very obvious to students what was going on when they were manipulating the physical objects.

Although neither of the graphs were linear it was useful to create them and to discuss what type of correlations there were and read information off the graph. We will revisit this concept again in the measurement unit. I look forward to seeing how well they retain the information.

As a side note, the graph of the maximum area was a parabola as expected. Without thinking too much about it, I expected the perimeter to do the same. As I saw students' graphs I wondered why they were the shape they were. I did the algebra and recalled that the resulting function was a rational function. Looks like a good problem for the grade 12 students. Tomorrow: Determine the function that minimizes the perimeter of a rectangle.

Let's Start Coding

The second week in December is Computer Science Education Week (CSEd Week). I get really excited about this because more teachers start talking about coding. The trouble with this event is that it often seems to be a one off event. Teachers give up an hour of their curriculum time to participate and once the hour is done they tend to move on. It's great that they participate but it could easily become part of any teacher's curriculum. I realize that it's "one more thing" to do but it can be very engaging for students and I believe that it really helps develop skills that are useful in many disciplines both in and out of the class.

Why are teachers not extending coding into their regular routines? I think for many of them it's about comfort. The Hour of Code tutorials are great. They are well laid out and could be coordinated by anyone. If you're going to start building coding into your classes, however,  you need to understand the tools a bit and you also have to find a way to weave in some curriculum expectations. No small task.

Brian Aspinall is a teacher who has not only embraced coding in his class but has taken it upon himself to help other teachers see how their students can code and meet curriculum expectations at the same time. He uses Scratch, which is really easy to use. You can find his videos here.

Brian's work had inspired me to play around more with Scratch and to find more ways to work it into the curriculum. I've decided to create videos that introduce some coding ideas but also create a challenge for teachers (or students) to work through. Hopefully, some teachers out there will decide to follow up on the challenges. If not, at the very least I will have thought more about how coding can be woven into my courses.

If you're interested in integrating computer science into your curriculum check out the #CSk8 hashtag.

Here's my first video. 

Trigonometric Regressions with Desmos

I decided that this year I was going to make trigonometric modelling a little easier for my students. I've used Kate Noak's Moon Safari in the past but I found that something was lost using a graphing calculator. The process of entering the data is time consuming and causes some students to get turned off the activity before they managed to get to the good stuff. This year I decided to give Desmos a try.

I entered the data into a Google spreadsheet that I made public. I gave students Chromebooks and had them open the spreadsheet. They copied the data from the spreadsheet, then pasted it into Desmos (yup just two steps) and then got to work trying to determine the equation that best modelled the situation. I really liked that they could see the graph as they modified the equation and see the coefficient of determination to help them determine if one equation was better than another. When they were done they could get Desmos to do the regression to see how close they were.

This was so much simpler than using a graphing calculator. It seems like Desmos gets better every time I use it. Thanks Team Desmos!

Dismissing Long Division

Every year I teach grade 12 students how to divide polynomials. I always start by reviewing long division with natural numbers. As I put an example on the board, I'm always met with groans and comments such as "I never learned this. I was taught it but I never learned it.". I always reassure students that it will be much easier now than when they were in grade 4. I'm blown away by how many students hate long division and these students are our best math students. If the majority of the best math students hate long division, what do the rest of the students think?

Every time I teach this lesson I can't help but think that students in grade 4 aren't really ready for long division. It's a long algorithm that likely makes little sense to them. Perhaps they aren't ready for it cognitively. I'd even go so far as to say that although many adults can perform the algorithm, how many of them can actually explain why they perform those steps?

I'd like to see long division scrapped from the elementary curriculum. Instead I'd like to see students focusing on understanding what division really means in a wide variety of contexts. Clearly it's a skill that my division-phobic students have not used since grade 4 so what's the point in teaching it then? Often when I mention this in conversations I get the reply "How will you teach them to divide polynomials if they can't do long division?". This argument seems incredibly weak to me. Does it make sense to teach students something in grade 4 and then ask them to recall it eight years later, without having used since grade 4? Why not just teach the algorithm in grade 12 when students have a better mathematical background and can understand what is being done rather than memorizing an algorithm that is likely to be forgotten?

As a side note, I really like James Tanton's representation of long division here. I think that conceptually it gives a better understanding of what is going on than the traditional algorithm.

What are your thoughts on long division?
 

Troubled Start

This year, for the first time, I was not at all excited to get back to school. I'm not talking about the traditional 'The end of summer is near. I don't want school to start' kind of apprehension. For whatever reason I was not interested in teaching this year. It seemed that I had nothing to look forward to. To put this in perspective I'm usually excited to get back to school and start teaching again. I've always found something to look forward to. This is what has allowed me to enjoy teaching for all the years.

What was different about this year? I spent a lot of time thinking about it and didn't come up with much. Was this the mid-career lull for me? Was I about to become an old crotchety teacher that didn't care anymore? I figured that once the fist few days were finished I'd be back in the groove. That didn't happen. Was the problem that I'm teaching the same courses and I'm happy with the start of those courses? Same old, same old?

It wasn't until midway through the second week that I may have stumbled on a possible explanation. I'm teaching a grade 9 course. I've never met most of these students before. My grade 11 class has a small number of students that I've taught before but I only taught them for a couple of month, two years ago, before I took a leave. Finally, I have not taught any of the grade twelves, in my class, in grades 9, 10 or 11. I really don't know any of my students very well.

It occurred to me that teaching is as much about building relationships as it is teaching content. It's about the student who is really good at math and finding a way to challenge her. Or discovering the student who is very capable but lacking self confidence and helping him develop that self confidence. It's about working with those students who need extra help on a regular basis and getting to come in for that help. And the list goes on and on.

Now that I have gotten to know my students, a little, I'm excited to help them reach their goals and help them be successful. I'm excited to further those relationships and help them (as well as myself) grow. Here's hoping for a great semester.

Connecting Globally vs. Connecting Locally

For the past number of years (three? five?) I've been amazed at how much learning and sharing I've been doing with math teachers from around world thanks to what's known at the MTBoS or the MathTwitterBlogoSphere. The MTBoS is essentially an online community of math teachers who share ideas and resources, bounce ideas off one another, etc.

During the time that I have been involved with this great online learning, I've found it very strange that I interact with math teachers from around the globe more than I do with teachers in my own district. This seems odd and totally wrong to me. How is it that teachers within the same district, within driving distance of each other, somehow don't have the opportunity to connect? How do you connect locally? Is it through organized professional development sessions or informal get-togethers after hours? I'd love to hear some strategies that work for you.

On a somewhat related note:

Last week while I was reading Dan Meyer's Public Relations post on appearances he has made, the feeling of oddness I mentioned above reared its head again.  Dan posted a link to the Ontario Ministry of Education's Leaders in Educational Thought: Special Edition on Mathematics. Despite the fact that I'm a math teacher, in Ontario, who is very interested in mathematics education and despite the fact that this material has been around since September 2013, this was the first time I had seen it. How is it that blogger from California is letting me know about resource that are available from the ministry that I work for? It's entirely possible that I missed some sort of memo or something but I would guess that I'm not alone. I feel as though somewhere along the line the message got lost. It seems a shame that money is being spent on these resources that don't seem to be getting used. Do you work as a teacher in Ontario? Did you know about these resources? If so how did you find out about them? I'm asking not to lay blame but simply out of curiosity.

I did a little digging and discovered that it's possible to subscribe, via email, to the Edugains site. By subscribing you will get an email update anytime new materials are added. To subscribe go to the desired section and click on the orange RSS button at the bottom. I also contacted the people at the Edugains site to see if there was a way of subscribing via RSS. It turns out that there is not at this time.

Students Dislike Independent Learning

When Young Children Hate School by wecometolearn

One of the biggest complaints that I get from students who come back from university to visit is that I don't force them to learn on their own enough. So today I tried to get my grade 12 Data Management class to learn on their own from the textbook. In doing so, I was reminded why I don't do it very often. Students hate it!

I started today by letting my class know about the feedback I have received from former students. We talked about how being able to learn on your own is a valuable skill. I provided a few pointers on how to pull out important information and then let them work.

I saw varied levels of participation. Some students were blindly copying definitions, others skipped right to the assigned work without reading and some didn't do much of anything.

Some of the comments I heard were:

"Can't you just teach us?"

"Why do I need to do this. I'm not taking math in university?"

"As I read this I don't remember anything."

I should point out that the content being covered wasn't difficult, which is why I chose to make this section independent. The barrier wasn't the content. It was the reading. Not that they can't read but that they'd rather not read.

I feel like reading to gather information is a valuable skill, in all aspects of life. I'd like to help my students become better at it, but today was very painful (for them and for me). What strategies do you use to help students get better at learning from a book?

Group Test

Last semester I taught the Grade 12 Advanced Functions course. It seemed that every time a test approached a student would ask if they could write the test as a class. We all had a good laugh then inevitably someone would ask if they could write in groups instead. Needless to say the entire class thought this would be a good idea. I dismissed the idea on a number of occasions explaining how it would be difficult to have a good sense of who knew what in a group. My students, however, were very persistent and would ask every time a test was nearing. 

On the second last test of the year (just before the Christmas holidays) a student asked if they could write their test as a group. I jokingly said "Sure" and a student immediately replied "Really?". When I told my students I was just kidding they provided a lot of reasons why such a test would be a good idea, in the hopes of getting me to change my mind. I let them know that I would think about it for a bit and get back to them; possibly a strategy for delivering a delayed "No".

As I thought about it I had a lot of questions about logistics for this possible test. They included:

1. What would such a test look like? Surely it couldn't be a regular test that students worked on in a group.

2. How will the groups be determined? Self-assigned? Teacher assigned?

3. How many students should be in a group?

4. What happens if some group members aren't pulling their weight?

5. Do students hand in one test each or one test as a group? Do they get the same mark or different marks?

6. Is this a bad way to prepare students for University?

Some of these questions and their possible solutions occupied my thoughts for several days before I had the courage to go ahead with it. I figured that if things didn't work out I could always call it a test review and give a traditional test afterwards.

Here are the answers that I came up with to the above questions.

1. The test should be less knowledge based (although there were still some knowledge questions) and should be more heavily focused on thinking and problem solving. The knowledge would show up as part of the problem solving.

2. I decided to let students choose their own groups and as it turns out students tended to group themselves by ability level, which is probably how I would have grouped them.

3. I went with three students in a group. I felt that this would allow for some good discussions while not allowing anyone to sit back and do nothing.

4. This is not that different from any other type of group work (assignment, presentation, etc.). The difference is that here I was able to watch to see who contributed what. It would have been possible for me to assign different marks based on the participation, which I didn't do.

5. Students handed in one test and received the same mark.

6. Perhaps, but it was only one test. Besides, is my goal to prepare students for university or for life beyond university? I would guess that once out of school most of these students will do far more collaborative work than they will test writing. Shouldn't I be preparing them for that as well?

Here are some things that I observed:

• There was no anxiety as students entered the class.
• There were some great discussions happening the entire time
• There was some learning going on during the test. Students who didn't understand didn't just let their group do the work, they were trying to understand it.
• There were no questions that were left blank.
• Students seemed to be enjoying the test.
• Students reported that the time just flew by.
• We had a modified schedule the day of the test. Our class was shorter than normal but I told the class that they were welcome to stay into lunch if they wanted to. Most stayed for the period and most of lunch. I was amazed that nobody just wanted to leave.

Here are a few comments that I heard during the test:

• After some discussion with the group..."I think I understand this now"
• S1:"That works!" S2: "Yeah it does." S3: "We've got it!"
• "YES! That's it."
• "I love this test. It's great to communicate."
• A student to me: "Can you tell me...?" Me: S:"Maybe I'll ask my group."

The test was a big hit among students. They said afterwards that they felt less stressed, they really enjoyed bouncing ideas off one another and wished that all tests could be done in the same way. From my point of view it was a great experience as well. Students were totally immersed in the work, there were lots of great discussions and the atmosphere in the class was very pleasant. It almost felt like a coffee shop, a productive coffee shop.

How did the students do? I would say that they performed at about the same level they normally would despite the test being more challenging than a typical test I would give. My hope is that by the end of the test they came away knowing more than had they written a regular test. I didn't measure this but I suppose a regular test after the fact might have provided some insight.

This is certainly something that I will try again. 

Student Voice

I'm a Sunflower by Mo Costandi

One of the big ideas in education now seems to be "student voice". My understanding is (and trust me I'm no expert) that if we allow students to voice their interest and allow them to follow those interests that they will be more engaged in their learning. I guess the thinking is that students have had a hand in deciding what to do and as a result are more likely to take ownership of their learning. I don't disagree with this idea in theory but I witnessed two incidents this week that make me think that the execution is often flawed.

Earlier in the week I attended a meeting of teachers and administrators. The purpose of the meeting was to discuss student engagement. As the meeting went on it seemed to focus on students voice. The thinking was if we let students learn about things that interest them they will be more engaged. As I said earlier, I don't disagree with the statement but I do think there are constraints in place that don't allow this idea to work. The two biggest constraints are likely time and a prescribed curriculum. On a smaller scale, one member of the group mentioned the fact that teachers are supposed to be posting learning goals for the day. How can a teacher post a learning goal if the students are deciding what they are learning about? The response from another member of the group was that the teacher could ask for ideas then steer the students toward the intended goal for the day. I'm certainly not an expert in student voice but it seems to me that by giving students a pseudo-voice the absolute best we can hope for is pseudo-engagement. This is a far cry from true engagement.

The second instance of student voice that came to my attention this week was in at my son's school. My son, who is in grade one, came home one day this week and told us they were going to learn about the sun. He said that the class shared all kinds of ideas they wanted to learn about with respect to the sun. I was pretty excited to hear this since he loves space and science. He wasn't excited. When I asked him why his response was that the teacher decided they were going to do an experiment where grow plants in different light conditions. I thought perhaps this was more pseudo-voice, but perhaps my son was just upset that his idea wasn't chosen. When my wife was at the school this week she happened to talk to another grade 1 teacher and discovered they were doing the same experiment. When my wife expressed surprise about both classes using their voice to come up with the same experiment the teacher's response was something along the lines of "Well, we kind of guide them to it". Pseudo-voice.

Don't get me wrong, I think the experiment is a great idea and I think my son would have really enjoyed it had the teacher said "This is what we're going to do". But I think there was some other topic that he would have rather explored and now he's disappointed the teacher is "making us do an experiment". It's amazing to me that evan at the age of 6 students are able to see what is going on.

Please note that I mean no disrespect to the teachers mentioned above. They are simply doing what is being asked of them while still working within their constraints.