Physical phenomena for quadratic relations

I’m working on a quadratics unit for my MCF3M (online) and MBF3C (F2F) classes. The Ms need to be able to do a few more things, but both groups have to be able to model quadratic “stuff” using an equation.

I’ll be using pretty heavily, and I got some great ideas from Heather Theijsmeijer (@HTheijsmeijer).

I’m trying to find some examples of physical phenomena that I can have students in either class play with to practise/demonstrate modelling. Here are my ideas so far:

Throwing or Bouncing a Ball

This is the first thing I thought of. A ball follows a nice parabolic path in the air if it’s moving horizontally.

My plan is to have students use a phone or camera to record a video or a rapid burst of images, overlay a set of axes, and fit a curve to the path. My iPhone can record at 120fps, which is great. I also found a handy post at Stack Overflow that explains how to extract images from a video, so that might be helpful too.

Pouring Water from a Hose

Set your hose at an angle, turn on the water, and snap a picture. Parabola. Beauty. Maybe put a piece of grid paper behind it, or just import it into Desmos.

Rolling a Ball Up An Incline

This one’s messy, but I think it might work.

Dip a marble in some ink or paint. Set a piece of grid chart paper on an incline (say, a piece of plywood) and roll the ball on an angle up the paper. When it crests and rolls back down, it should have left parabolic paint. On graph paper.

Other ideas?

I’m open to suggestions. I have stuff like photos of suspensions bridges, etc., but I really want something students can generate on their own.


Key Learnings from our e-Learning Collaborative Inquiry

I learned a lot over the past couple of days. My board brought in Donna Fry (@fryed) and Tim Robinson (@timrobinsonj) to guide our group’s learning, and several folks from the board’s central program team were helping out as well. Check out #elADSB for a bunch of new Twitterers, too.

Here are some prompts that we were asked to respond to, and here are my (rather brief) thoughts.

1. ​​What questions, wonderings do you have ​with respect to our problem of practice?

How do we encourage collaboration that isn’t false collaboration? How do we ensure that the collaboration is meaningful and valuable? I don’t want to force collaboration among students when it doesn’t make sense.

What does a rich task look like in a course which has a lot of technical, procedural learning? Is it enough that there is inquiry in the task, or does it need to be “authentic” and relevant? I’m concerned about fauxthenticity: forcing an unrealistic application out of a concept just so that we can say it’s “real-world”.

2. What key learnings have you had over yesterday and today?

I need to provide more structure for self-monitoring and self-reflection for my students, and then ensure that they follow through on that self-assessment. They’re still kids, and they need a firmer hand with organization and checking on their learning, or they may neglect important stuff (or even delude themselves into thinking they understand concepts that they don’t).

I need to consider including more face time in my course – maybe having “Math Chat With Mr. G” or something on certain afternoons… Even if some students can’t get me live, I shouldn’t prevent it for those who can.

3. Moving forward, as a result of your learning, how do you envision your courses/practice changing?

I need to be more careful about connecting students with each other instead of encouraging them to rely on me.

I need to set up a fast, reliable, easy-for-me-to-check-in-on self-monitoring system for my students.

I will set up Adobe Connect or Google Hangout time periodically.

I’m going to work on figuring out when requiring collaboration makes sense, when students should be working independently, and when they just have a choice (most often?).

I’m going to make some plans and then ask someone else to give me suggestions about them before I implement them. :)

LaTeX Math for e-Learning in D2L

I am teaching MCF3M online this semester, so I need to be able to include math notation in my online content, quizzes, etc. I know how to write math notation using LaTeX from my days at the University of Waterloo, and I find it a lot faster than using a graphical equation editor. I’ve tried Microsoft Word’s editor, which accepts LaTeX-like input as well as graphical input, but I still find it frustrating to use.
I’m teaching in the Desire2Learn/BrightSpace learning environment, so I need to ensure my content works well in there. Last semester I taught Computer Science/Programming and used PDF files that I created in Word Online, and I considered doing the same thing again.

But D2L has an equation editor as part of its HTML editor for webpages, discussion posts, etc. Could it be all I need?

I’ve taken it for a spin before. Here’s the workflow:

Create a new page and type into the HTML editor.


Expand the toolbar so that the Equation tools are available.


Choose \∑ LaTeX equation.


Type in the LaTeX expression, using \( and \) as delimiters for inline mode (otherwise it defaults to block mode).



Looks good.


But look at the source HTML code:


Uh-oh… that’s MathML (Math Markup Language), not LaTeX. What if I want to change something in my original LaTeX?

Well, you can see at the bottom that my LaTeX code is still there, but it’s not being used. I could remove all the MathML, cut out my LaTeX, modify it, and re-insert it using the LaTeX equation editor.


I thought that MathJax, the rendering engine that D2L uses for math notation, could only handle MathML (since notation from both LaTeX and graphical editors are converted to MathML), but it turns out that’s not true. MathJax can do LaTeX.

So I tried putting LaTeX directly into the WYSIWYG editor:


No dice.


The trouble is that D2L has parameters on its JavaScript call to MathJax:


That config=MML_HTMLorMML bit is saying that only MathML is acceptable input (and HTML or MathML can be output).

So I added another call directly to MathJax in my own source code:


I set the parameter to be config=TeX-AMS_HTML, which will accept my LaTeX input and render in HTML/JavaScript.



But this is kind of a pain.

I can use D2L’s editor to insert math, but I get MathML (which I find hard to edit).

I can write in LaTeX and have it be preserved, but I need to add a script call to the start of the HTML source code (a hassle, but not too serious, I suppose).

Or I can write in some other (offline) development environment, include my script call all the time, and just upload my completed HTML files to my course. This has the advantages of being independent of D2L, available without internet access, and very shareable.

So that’s what I’ve decided to do, at least for now. So I’ve learned a little CSS to make my pages less vanilla/more functional, and I’ll try to improve the look and feel as the semester progresses.

Wish me luck.

My students like the class blog (phew!)

I reminded my MDM4U students to visit our class blog today as I had posted a video this morning with solutions to some questions from class. I was pleased to hear that a few students have been heading there regularly. I played a bit of the video on the screen so they would understand what it was like, and then I asked for input about the sorts of resources they would like me to post.

First, most students want to have the videos. They’re nothing magical; they’re just me solving some problems on paper. A couple of students said they like the idea of being able to rewind me (since that’s hard to do live, in class).

Second, a few students also want a text format of some kind (a blog post, a PDF file, or something like that). I understand that; it’s faster if you just need a short bit of explanation. It takes a lot longer to type out a solution, but I could always write one and take a photo, I suppose.

The third response I took away was that posting videos like this is unusual in their experience, and they appreciate the extra effort on my part. Several students seemed to understand how much extra work goes into creating these supplementary resources, so that was nice.

At some point I’m hoping to post more student work for them to learn from instead of creating everything myself. I think they’ll learn a lot from doing it, and then a lot from each other. Videos would be awesome…

I care a lot about their success, and I’m hoping to give my students a better-than-average chance at being great at this stuff. A lot of the responsibility lies with them, but I’ll remove as many barriers to their learning as I can. I’m glad to hear that my work on the blog is being well-received.

Teachers DO care, Seth Godin

Seth Godin wrote a blog post yesterday which has been making the rounds in education circles. It’s called “Good at math”, and in it Seth makes a very good point in the first paragraph:

It’s tempting to fall into the trap of believing that being good at math is a genetic predisposition, as it lets us off the hook. The truth is, with few rare exceptions, all of us are capable of being good at math.

He’s right, without question. Most people are capable of becoming “good at math” the way most people are capable of becoming “good at reading” or almost anything else.

The problem with math education, he argues, is that there has been a focus on memorizing formulae in math classes instead of on the important stuff (I’m assuming he means modelling, problem solving, critical thinking, etc.). In particular, he says standardized tests are useless.

Before you continue, please understand that I agree with Seth. I agree that math education has been unbalanced in favour of rote learning and memorization, sometimes to the exclusion of problem solving and critical thinking, and is divorced from real-world, relevant application.

But one sentence in his post is really galling to me. One point he makes, almost casually, that has prompted this post.

He says,

It’s because you haven’t had a math teacher who cared enough to teach you math.

Again, I agree that the focus has been wrong, and that math education has been largely ineffective. But that’s not for lack of caring on the part of teachers.

Teachers are among the most caring people on the planet. We work like dogs, struggling beside our students, coaching, cajoling, encouraging, pleading. We can’t sleep for thinking about how to improve for the next day. Teachers agonize over their students misunderstandings, apathy, home lives, social problems, mental state, self-esteem, literacy, and all other aspects of their beings. We are charged with developing tomorrow’s citizens and we take it very seriously.

But it’s true that not all teachers are exemplary. Not every class is equally “engaging”. Teachers don’t all have the same arsenal of strategies at their disposal. We disagree about assessment, behaviour management, instructional paradigms, big ideas, learning goals,… pretty much everything, actually. There is little consensus among teachers.

Except for one thing: we care. We want the best for our students. We want to be better at our work.

So why has ineffective math education persisted for so long?

  • there is a cycle of successful math students becoming math teachers (this is true in every subject area, not just math)
  • it’s hard to employ new strategies
  • it’s hard to find mentors who have effective practices and can coach
  • there are approximately a bazillion other demands which take away from instructional practice, including other system- and school-level priorities

That’s a lot of reasons/excuses, but there are solutions.

Teachers must become aware that there are ways to improve their instructional practice. Some folks believe that the hard work is figuring out how to ensure compliance from students. “If only they would [insert good learning behaviour] then they would be successful” is a common statement. Developing this awareness requires exposure to better methods, whether through direct observation or other sharing.

Teachers need time and “permission” to try strategies which are new to them. Fitting everything into a math course is really challenging. Now ask a teacher to “take chances” with several lessons. It’s hard to convince people to try something new when you can’t promise it’ll work. In my experience, the best way to make the case is to remind them that what they’re doing now is only working for their strongest students anyway. That time spent “experimenting” can be gained back in the form of increased student understanding and skill.

School leaders need to encourage and support teacher learning in dramatic and meaningful ways. Make co-planning a priority. Visit classrooms. Learn about effective math instruction so that you can have rich conversations with your math teachers. Finding the time to develop as a school leader is possibly the most difficult task I’ve mentioned.

Seth’s blog post seems to have been aimed at the general public, reminding us that numeracy is as crucial as literacy, and that we have been damaged by narrow, ineffective teaching practices. That is a great message, and one that I think most people are hearing in his words. His claim that it’s because teachers don’t care is ridiculous.

Student questions about probability

After yesterday’s realization that I was directly the flow of learning too much in my class, I asked my students today to generate some questions they were interested in regarding probability. Here are their responses (posted also on the class blog at

  1. What are the chances of winning the lottery?
  2. What are the chances of finding a shiny Pokémon?
  3. What are the chances of the earth being destroyed by an asteroid?
  4. What are the odds it will snow tomorrow?
  5. What are the chances of winning a car in Roll Up The Rim?
  6. What are the chances of being “caller number 5” on a radio contest?
  7. What are the chances of your seat being picked for the million-dollar shot at a basketball game?
  8. What are the chances of being struck by lightning?
  9. What are the chances of finding a $100 bill on the ground?
  10. What are the chances of getting all red lights on the way to work?
  11. What is the probability that a solar storm wipes out Earth’s electronics?
  12. What are the odds of an average poker hand winning?

I’m proud of their questions. I can see that some of them will be very difficult to answer, and others fairly easy. All of them will require some thinking about possible outcomes or statistical probability (which we haven’t studied yet, so that’s pretty awesome).

Tomorrow we’re going to start trying to solve these questions. I’ll give the students the list, and we’ll start drumming up solutions in groups using chart paper to record thinking. I’m pretty excited; I hope they are too. There is a ton of excellent learning that can come out of this.

I made another mistake: missing out on inquiry and authenticity

I’m teaching MDM4U (Data Management) this semester and we’re starting to talk about probability. We’ve spent the last few weeks learning a bunch of counting techniques (permutations, anyone?) and soon we’ll be applying those techniques in this new context.

But I’m concerned about how teacher-directed everything has become, and how comfortable my students seem to be with that mode. When does their curiosity take control of our journey? How will their interests drive our learning?

On the first day of the probability section I was speaking with the entire class about the sorts of probabilities they would be familiar with: chance of rain, poker, winning a football game, etc. One student asked, “What are the chances of winning the lottery?”

And I made a big mistake.

I told him, “We’re going to look at that when we have a few more tools to work with.”

I should have said, “Let’s try to figure that out. Now.”

His curiosity would certainly have driven him and other students to pursue an answer to that question. No, they don’t necessarily have the skills to answer that yet (some would), but I also don’t need to teach a bunch of lessons before they can start.

I should have encouraged him to frame that question mathematically, identify the information that would be needed to solve it, and begin to do so.

Instead I put him off and went on with my boring talk about rolling dice and flipping coins. I missed a great opportunity for authentic learning in favour of simple, canned questions.

So, my deepest apologies to that young man and to the rest of the class. Tomorrow, I fix it. Tomorrow, you will decide what you want to learn, and then you’ll learn it, and I’ll be there to coach you along the way.