Posts tagged math
No, I'm Not Listening
Students say the best things when they think I’m not listening.  It’s pretty funny actually, if not more than a little frustrating.  Students can be having excellent conversations about mathematics but as soon as I come over to listen they get just a little bit self-conscious and the conversation ends.


I’ve gotten really good at listening from far away.


We’ve been dabbling in project based learning with our grade nine academic classes this semester.  After our first project, we introduced our students to Dandy Candies.  This activity has been around for awhile.  I first was introduced to it at the 2015 OAME conference when Dan Meyer ran it with a packed room.  Since then, I’ve used this activity on and off with my classes, but never as a project.


The premise is pretty simple, show students the video of a preset number of small cubes assembling into packages (prisms) of various dimensions.  Then ask them some questions about what they’ve seen.


“What do you notice?”
“What do you wonder?”


As the person responsible for steering the conversation, I try my best to guide them to the questions I really want them to ask.  Except this time my thunder was stolen!  Apparently, after doing only one previous project, at least one student in the room was wise to my plan.


“I bet Mr. Mitchell wants us to figure out which arrangement uses the least wrapping paper”, said one student.  The entire classroom kind of gasped and went quiet.  Then there was an explosion of conversation at each table group.  This is a win.  Here I thought I’d have to trick my class into asking the right questions.  Somehow, they did it for me.




We spent a week playing with Math Cubes, calculating surface area and preparing a final report on which arrangement of 48 and 128 cube sized candies would need the least wrapping paper or in math speak, had the least surface area.  Then we spent the following week diving deeper into Surface Area and Volume calculations.  Students showed us their understanding in different ways.  It was fun!




It has now been 7 weeks of project based learning and curriculum wise, the grade nine team was all over the map, covering linear relations and geometry without having developed some of the other tools that would make solving these problems easier.  This was on purpose, we were creating a need in our students for them to ask us for better tools.


Up to this point, I haven’t once mentioned the word algebra to my class.  But it was time to introduce them to some more formal methods of problem solving.  Todd Malarczuk provided us with a simple mini project to drive home the need for algebra.




Students were presented with some pictorial representations of different algebraic equations.  You’ve probably seen these puzzles floating around your Facebook feed.  You may have even witnessed some of the arguments that they create.  Well, it wasn’t surprising that the same thing happened with our students, arguments everywhere!  Yet as I roamed around the room observing how the students were working, listening but not listening, I heard some comments that convinced me this was working.


“Drawing apples in my notes isn’t very easy.”
“If a bushel of bananas is four, do we need to know what one banana equals?”
“I wish there was an easier way to do this”

We had created a need in our students for something more efficient.  They wanted to know how to calculate what a coconut equaled, without us having to convince them that they needed algebra.  It was difficult, but by giving students the time to struggle; to try and fail and learn, we had created a community where risk taking was valued.  It wasn’t taboo to ask questions about the math we were learning or wonder about where we were headed.  Students owned their own learning, without us having to tell them that the learning was important.
From Why to Barbie Bungee
Back when I started down this ever Shifting road, I learned that good Innovation starts with identifying a WHY.  Why might I want to let students access technology in class?  Why is building a positive classroom culture so important?  I spent the summer thinking about the direction I wanted to take my classes this semester and why I wanted to take them that way.  Neil deGrasse Tyson summed up my thinking pretty well.



I’ve noticed that try as I might, the students I encounter lack a spirit of creativity.  It seems that many times they would prefer to be presented with the answer rather than seek out an answer on their own.  Thinking is difficult.  Many students say they enjoy learning, but that the stress of grades and assessments turn passion into anxiety.


I want to change that.


In May, I was fortunate enough to be able to attend the Ontario Association of Mathematics Educators (OAME) annual conference in Kingston.  I saw lots of great pedagogy that has been put into action and enjoyed many great conversations about how we can move our practice forward.  I felt empowered to try something new and with the course team at my school, we decided to forge ahead with some project based learning in our Grade 9 Academic Math classes.


The course team decided to start with a week of community building, because nothing pumps the brakes harder on curiosity than a week of prerequisite skills worksheets followed by a quiz.  We had fun and laughed by playing team pictionary on our first day of class.  We did some problem solving activities around the game Skyscrapers.  We debated how tasty a banana really is by playing around with Desmos Pomegraphit.  We ended the week by playing Battleship and talking about situations when plotting points might be important.


All in all, in four days we didn’t do much traditional math but the class was primed for our first project, Barbie Bungee.  This is a project that has made the rounds with math teachers for years.  I had heard about it, but never had the chance to try it with a class of my own.


The premise is simple: tie an elastic around Barbie’s legs, drop her, measure how far she drops, add an elastic and repeat.  Then figure out how many elastics are needed for a very high drop and try to get Barbie as close to the ground as possible.


On Day 1 we started with a Desmos Activity followed by a teaser video of some Barbie’s not having much fun.  




We polled the class with various questions: What do you notice?  What do you wonder?  Doing this allowed us to slowly introduce the project and steer the class into asking the real questions:  What makes the “best” Barbie Bungee?  We ended class by defining the “best” bungee as being the one that takes Barbie as close to the ground as possible without physically touching the ground.


On Day 2 we pushed forward by asking students to sketch what a relationship between the number of elastics and the distance Barbie falls might look like.  This prompted some good discussion on what we expected to happen with our Barbies as we added more elastics.  Then it was time for the actual challenge to start.  Without leaving the classroom, students had to build bungee cords for their dolls so that they could be safely dropped from the top of the doorframe.  It was a bit of a disaster, but the teaching team expected that.  Our students hadn’t really been shown the tools to be successful at this activity.  We were, to borrow a phrase, “creating a headache” so that students would feel the need for the math.





After watching our students attempt this initial challenge with no real plans, we pushed pause on Day 3 and did a more formal type of lesson.  We reminded our students about Scatter Plots, and discussed some ways they could use this concept to help them with their Barbies.


No Barbies were harmed...
Given this knowledge, we started Day 4 with a better plan.  Students knew to start small, measure the distance Barbie falls with one band, then two bands, then three and try to find a pattern.  We wanted our students predicting, given that we had not yet told them the real height of our Bungee Jump.  By the end of the period all groups were ready to test their Barbies and were told that the drop on the last day of this activity would be 520cm, or 5.2 meters, or two stories.


Day 5 was the big day, a time to test our students calculations in the real world.  There was a lot of excitement and smiles.  Perhaps not surprisingly, there wasn’t much anxiety.  This was, for all intents and purposes, a test of the students work and knowledge over the past week, yet none of our students approached it this way.  They were eager to have their Barbie’s jump, and disappointed when we told them they couldn’t refine their bungee cord after their failure.  And there were many failures.  

The point of this project wasn’t really to build the “best” Barbie Bungee.  The point was to get kids excited about math class, to get them to think about math class, and to let them have some fun in math class.





At the end of the week the students had learned a lot by doing.  They’ve explored scatter plots, collected data, and made informal lines of best fit.  They were assigned to visibly random groups at the beginning of the week and met people they wouldn’t have otherwise worked with.  


Could my class do well on a formal test about scatter plots and lines of best fit?  Probably not at this point in the course, but this is why I reject testing as a sound form of evaluation.  When a student completes a test and fails, what have they learned?  When students dropped their Barbie’s and they hit the floor, what have they learned?  I’d argue that students that have iterated their way through the bungee jump project have learned much more from their own failures than those who fail a test.  From my point of view we brought some joy into our curriculum and lets students learn through a bit of discovery.  It’s working for us so far!

So that’s my why, what’s yours?
Teaching and Learning with Desmos
Me and my math classmate and helper Nikolina
The other day, Jamie asked whether I wanted to become a student again for a math period.  All for it, I grabbed my laptop (as I was told I needed it), and ran for the portable.  You see, Jamie was doing a little recording of Stephanie Briggs’ Grade 10 math class for the upcoming OAME (Ontario Association for Mathematics Education) conference, where Briggs will be presenting on teaching using Desmos with Todd Malarczuk. I had heard of Desmos, the online graphing calculator, because well, Jamie won’t ever stop talking about it!  As an art teacher, I never really knew what he was always so excited when Desmos was mentioned. So here was my chance to see what all the fuss was about.


I settled into a group, hoping that my new math classmates might help guide me through the period.  I have to say, it is a little intimidating diving back into high school math, not having done so SINCE high school *cough* a few years ago.  Briggs introduced the day’s topic quadratic equations in vertex form and did a quick review of standard form and factored form.  I calmly wrote down the equations that she was putting on the board and pretended to follow along.  Really though, I was panicking a little because right from the top, I was lost.  I didn’t know what the formulas were supposed to do, and even what all the letters in the formula meant.  I sidled up to my new math pal, Nikolina, praying that she could help, and she was great.  She quietly explained the formulas we had just written down.


Next, Desmos!  Students seemed excited to get started and they were ready, devices in hand.  We were doing “marble slides” , where you were trying to create the shape of the parabola so that marbles slide down the parabola and pass through stars.  Kind of a like a phone game, but with MATH! The only problem was, I had no idea how to manipulate the formula to alter the shape of the parabola, and we weren’t given specific instructions on what to do.  Frozen, and unsure how to proceed, it wasn’t until I confirmed with my seatmates that it was ok to just start stabbing in numbers and see what happened.  Trial and error, look for patterns, trial and error, look for patterns...  Try, fail, learn. Try, fail, learn.  Now I was starting to get it!  A little journey of self-discovery, which is why Briggs didn’t tell us what to do.  Brilliant!  Slowly I figured out what would happen when I changed the numbers and how it affected the parabola.
desmos_marbleslides.jpg
Then, all of a sudden, Stephanie Briggs paused the activity for all of us.  We were locked out! There was a chorus of collective groans from the students because they were so engaged and invested in what they were doing! With the class’ full attention again, she did a quick revisit of the formulas we were using, asking about what we had learned so far about how the numbers in the formula affected the parabola.  It was an interesting and effective use of a pause in the activity!  After consolidating our thoughts on the process so far, gleaning a bit more understanding in the process, she un-paused the activity, allowing us to get back into sending marbles through our parabola slides.  I was getting the hang of this, starting to gain confidence and have fun with it.  Not bad progress seeing as I was blindly stabbing in numbers only a few minutes earlier. Briggs was constantly circulating through the room checking in with each group individually during the activity.  The whole goal of the activity was for us to discover and develop what vertex form was, which for me and students at my table group, seemed to be sticking a lot more than if we were just told what vertex form was.  

Stephanie Briggs circulating and helping students through the Desmos activity
All in all, it was an insightful window into Math exploration.  I found it a little like the math equivalent of getting a new type of paints that I’d never used before and just going for it and seeing what happens.  I don’t recall that kind of open ended exploration and play in my math classes.  Students are definitely better for it.  Thanks to Ms. Briggs for the lesson, now I can see why they get so giggly and excited every time Desmos is mentioned!


If you want to play Marbleslides yourself you can go here.
If you feel like exploring some of the other Activities provided by Desmos, head over here.
If you want some Desmos activities, organized by course, to use in your class, head over here.