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.

PD Day plan: Meeting with my Math department

Tomorrow is PD Day in Algoma, and I have a couple of hours to work with the math department in my school (which I newly lead). I want to make the most of our two hours together, so we’re going to be spending our time [mostly] talking about assessment.

I’ve been working here for a few weeks, so I have some idea of the nature of assessment in each teacher’s class, and in the school as a whole. Some of that comes from talking casually with teachers in our shared office; some of that comes from my students explaining what they’re used to (common practices).

I have learned something in the last few years: hardly anyone has had the time I’ve had to read about, hear about, reflect upon, and discuss their teaching practice. That’s not a criticism, of course; I’ve just had the luxury of working centrally for six years. That’s a lot of workshops, a lot of meetings, and lot of one-on-one conversations with classroom teachers from all over the board.

I have thoughts about what good assessment looks like, so I could just tell everyone how I think it should be, but I know that’s not effective. First, I could be up in my sleep, having not practiced all of the strategies I believe will work. Second, people need to own their approaches, not just follow someone else’s.

A principal told me last year that school boards often make the mistake of having senior administration learn a lot so that they can make a decision about a system-wide approach to a problem. Then the school principals are “trained” or “in-serviced” or otherwise told how to implement this approach. But almost never is there really an opportunity for the principals to become deeply familiar with the solution (or even the problem!) in order to believe it’s the right choice.

So I’m trying to be careful to not make that mistake. I’m trying to coach in the way I know is best: encourage the learner to reflect upon the current practice, to question its efficacy, and to consider something else that has reason and research to support it.

The Plan for tomorrow

We’re going to develop a Working Agreement for our meeting (lots of folks use the term “norms”; I first heard “working agreement” and I like how it feels more collaborative than imposed). I don’t know how long this will take, but it’s worth taking the time now.

Then I have some reflection questions for everyone. I’m still deciding on strategies here (for practical reasons; there could be a dozen people in the room). These are the questions I’m considering:

  • What kinds of assessment do you use in your classroom?
  • What is the purpose of each kind of assessment you use?
  • How does each kind of assessment help students to improve?
  • How are students involved in assessment?
  • How do you record assessment information?
  • What are the rules/policies about assessment and evaluation (department, school, board, Ministry)?

There is a lot of background knowledge that goes into assessment, and I don’t yet know how common that knowledge is. I’m thinking about

  • Assessment For, As, and Of Learning
  • Conversations, Observations, and Products
  • Big Ideas, Learning Goals, Success Criteria
  • Summative Tasks, Richness, Authenticity

I have about 14 hours until we meet as a group. Any suggestions are very welcome.

LaTeX in WordPress

I assumed I would have to pay fees, get a plugin, or use in order to have math rendered in my blog posts. Not so.

Details are at, but here’s a sample:


Nice, eh? It renders an image and puts the \LaTeX in the ALT tag. I love it.

Math Problem: Better buy on cheese

You’re in the grocery store on April 24th and you need cheddar, desperately. Upon reaching the back of the store you discover that there is a sale!

Excitedly, you search the shelves and find a 500g package of old, light cheddar (your favourite) for only $3.99. Beside it is a 340g package of the same cheese; it’s $2.99.

Looks simple at first, but then you see the expiration dates: May 3rd for the 500g package and May 19th for the 340g package.

Last time you bought the 500g package it took your family two weeks to eat it.

What should you buy?

My answer for “How many area codes does Canada really need?”

I posed a question a few days ago: “How many area codes do we need in Canada?” Here are some of my thoughts. I don’t think this is a complete answer (you’ll see why at the end), but I do think it’s a good back-of-the-napkin attempt. I also think it’s worth considering this problem from the point of view of younger students, who won’t have a lot of math in their pockets yet.

Canada has about 33.5 million people. How many phones should each person have? Well, some of those people are children, some are parts of families, some work, some have cell phones… it’s not simple, is it?

Well, let’s assume most children under 10 years of age don’t need a phone to themselves (I sincerely hope that’s true). I estimate (but haven’t bothered confirming) that there are about 5 million such children in Canada. That leaves 30 million potential phone owners.

Let’s suppose that every one of those individuals has a personal phone (like a cell phone), and that every one of them also has an organization phone (e.g. for a business, church, club, etc.). This isn’t realistic, since most churches don’t have a phone for each member, but also most church-going folks also work, so I’m hoping for some balance here.

That’s a ballpark of around 60 million phone numbers so far.

Also, I’ll guess there are something like 15-20 million families who probably have another shared phone line (like a “landline”), so we’re rounding upwards to 80 million.

Okay, let’s step back and consider the numbers themselves.

Looking at the last 7 digits, they’re broken into the exchange (3 digits) and the end part (4 digits – see my technical jargon there?).

Exchanges can’t start with 0 or 1, so that leaves 800 possible exchanges. The end parts can be anything, so there are 10000 possibilities there. Multiplying those two values gives us 8000000 (8 million) phone numbers per area code.

(Yes, there are a few other exchanges you can’t have, like 555 and so on. They’re small potatoes in this calculation. I’m sure my rounding up other stuff overwhelms them).

All right, so now we have 8 million numbers per area code and a need for 80 million numbers. That means we can have 80 million / 8 million = 10 area codes, right? Hey, that’s just about one per province/territory!

Except the population isn’t evenly distributed across the country. There are 12.8 million people in Ontario and 3.6 million in Alberta. How to resolve this?

Well, let’s round our 80 million up some more to make the numbers nicer. Let’s say we want to future-proof this a bit in case there’s a sudden population increase. Let’s bump it up to 3 phone numbers per person, which is just north of 100 million numbers total. Then we can distribute numbers based on the populations in each region, taking care to not let area codes cross provincial/territorial boundaries for convenience (thank you to Wikipedia for the numbers):

Province/Territory Population Phone Numbers Needed
Ontario 12851821 38555463
Quebec 7903001 23709003
British Columbia 4400057 13200171
Alberta 3645257 10935771
Manitoba 1208268 3624804
Saskatchewan 1033381 3100143
Nova Scotia 921727 2765181
New Brunswick 751171 2253513
Newfoundland and Labrador 514536 1543608
Prince Edward Island 140204 420612
Northwest Territories 41462 124386
Yukon 33897 101691
Nunavut 31906 95718

Okay, so that leaves Ontario with 5 area codes, Quebec with 3, BC with 2, etc., until we arrive at a minimum of 21 area codes for the country. If I’ve counted correctly there are currently 37 in use, which is quite a bit more than my napkinning would suggest is necessary.

Of course, there are other factors I haven’t considered here. How many devices have phone numbers attached that are not for humans talk with? For example, what about automated calling systems? How many people acquire new phone numbers in a given time period? It would be nice to prevent numbers from being reused for a long time (say, a year), which would increase amount of available phone number space needed.

There are probably some other things I haven’t considered – anything come to mind?

Math Problem: How many area codes do we need in Canada?

A photo of the top of a telephone booth showing the word TELEPHONE

via wintersixfour at

I was thinking about this recently while going through session proposals for On The Rise. Presenters gave contact information, including phone numbers, when submitting their proposals. I noticed quite a few area codes in there.

In my area we’re part of the geographically massive 705 area code, but we acquired another, overlapping area code (249) last year. I haven’t heard of it being used yet, but we’re now on 10-digit dialing. I had a friend who lived in a small community near Waterloo, Ontario, who said that they were on 5-digit dialing for a very long time, into the 1990s, I believe. In my own community all of the phone numbers are of the form 705-248-****.

So, geography definitely has informed the distribution of area codes and exchanges (I believe that’s what the next three digits are called) because of the wired phone lines of the past. I imagine that the need for that kind of segregation of codes is technically past, although it’s still nice to know that someone calling from a 519 area code is based in Southwestern Ontario (although they could be next door on a cell phone).

Here’s the math problem

“How many area codes do we need in Canada?”

This question could be posed at a variety of grade levels. I think grade 4 or 5 students could handle the more basic parts of the problem, while it’s still very interesting for grade 12 Data Management students. A related (but surprisingly different) question is “How many phone numbers do we need in Canada?”

I will give my answers in another post, but maybe you can think about it. If you have a class of students, try asking them. Record the thinking, and post/link to it in the comments. I bet you’ll be surprised at the complexity of the questions and the richness of the answers.

A Sample Proof Using Mathematical Induction (playing with LaTeX)

It’s been a long time since I used LaTeX regularly, and I discovered that I don’t have any leftover files from my days as a math student in Waterloo. After looking at LaTeX in the context of D2L today, I dug out a unit plan I had written for MGA4U (extinct) in 2003 using the Ontario Curriculum Unit Planner (also extinct).

First it’s worth noting that I had no idea what good assessment looked like. It’s almost 11 years ago, but it feels like a lifetime when I look at the awful way I was planning to assess students. This was a unit I developed while practice teaching. I’m really, really glad my practice doesn’t look like that now.

Anyway, I found a page (“S2 Teacher Resource 1.pdf”) which I had typeset using LaTeX. I thought I’d try to reproduce the mathematical proof on the page to see if I could remember how. Here’s a picture of the page:

A photo of a sample proof on paper

I left off the trimmings and tried to write the middle. I can’t remember how to indent things the way I did before, so I’m settling for something different (feel free to educate me, though).

I used the JaxEdit LaTeX Editor at to create the document:

\title{Sample Proof Using Mathematical Induction}
\author{Brandon Grasley}

\\For any $n \in \mathbb{N}$, 
\[$\sum_{i=1}^{n}i=\frac{n\left ( n+1 \right )}{2}$\]
\\Base case $n=1$: If $n=1$, the left side is 1 and the right side is $\frac{1\left ( 2\right )}{2}=1$.
So, the theorem holds when $n=1$.
Inductive hypothesis: Suppose the theorem holds for all values of $n$ up to some $k$, $k \geq 1$.
Inductive step: Let $n=k+1$. Then our left side is
$\sum_{i=1}^{k+1}i&=\left (k+1\right )+\sum_{i=1}^{k}i\\
&=\left (k+1\right )+\frac{k\left ( k+1 \right )}{2}$\text{, by our inductive hypothesis}\\
$&=\frac{2\left (k+1 \right )}{2}+\frac{k\left (k+1 \right )}{2}\\
&=\frac{2\left (k+1 \right )+k\left (k+1 \right )}{2}\\
&=\frac{\left (k+1 \right )\left (k+2\right )}{2}$
which is our right side. So, the theorem holds for $n=k+1$. 
By the principle of mathematical induction, the theorem holds for all $n \in \mathbb{N}$.


This generated a page like this:

An image of the proof rendered from LaTeX


I’m definitely surprised at how hard it is to find free, online LaTeX renderers. I ended up taking screen shots of that output in order to post it here. The others I found either require sign up or didn’t successfully render all of the math components.

Well, something to keep exploring. I wonder if I’ll start working in HTML with MathJax instead, using D2L as the authoring platform. We’ll see.