Thursday, December 18, 2014

Slow Down...and Do the Math (Part 1)

The full text of the problem is missing, but you get the idea- it's a fairly typical seeming growing dot pattern.  Many of these patterns are deceptively simple- it's fairly easy to describe how it grows, to construct a table, and to come up with an additive relationship for how it grows.

From the Ontario curriculum, you might expect any student from even grade 2 and up to be able to say SOMETHING about this pattern.  The problem strongly fits explicit teaching of the math process "Representing"- tables, graphs, explanations, and algebraic representations all work.  The patterning/algebra curriculum is ideal for showing the variety of mathematical representations that we can use.

This problem worked very well for co-teaching in a grade 7 class.  After about 15 minutes of paired work, all student pairs had something down on the page.  They had made tables, extended the patterns, and made conjectures about how the pattern grew.

This is where I had a conversation with one of the teachers in the room about how the students "should" have been done 5 minutes prior. On the surface, that seemed like it might be true-there was something on every page, and the students' initial line of thinking was tapped out.

Having a bunch of experience seeing students work with growing patterns, I thought that many were "ready" to make the leap from additive to multiplicative reasoning, and quite possibly to generalizing using algebra for the nth term. Asking one or two key questions here would be the key to getting further with our thinking.

Here is an example of a pair that was ready to jump to generalizing:

The proof is the circling of the groups of 3 on the right.  Groups of 3 leads to the idea of multiplication by 3, which can get us to the ideas of variables and constants.

Another teacher suggested asking for the 15th term.  What typically happens here is that those students who need to keep adding to show the constant growth will mechanically extend the pattern, and get the correct answer.  But those who are ready to take the conceptual leap will often jump in with insights, like lightning out of a blue sky.

Here is where the differentiation happens- some students realize they understand the problem right down, all the way to the bottom.  They have broken it down to its mathematical elements, and can use those elements to construct their own explanations for what's happening.

Given an extra 5 minutes, here is what one pair had to say about the 15th term:

-it has 15+16+17 dots.

How did they know?  Term 1 has 1 +2 +3, term 2 has 2+3+4, etc. This was quite an efficient way of thinking about the pattern.  It would allow them to give the number of dots in the nth term with simple addition.

In the consolidation, some students revealed that they knew 3n + 3 would work for any term.  They  had little to no prior experience with algebra, but were able to make that generalization.  I think further work in this classroom (on other situations/patterns) could focus on generalizing through multiplying, and generalizing for any term.  Here is where precise explicit teaching through mini lessons, and purposeful practice work would come in.  The next week or two of classes could all be planned from what we learned that day. Such is the power of consolidating our thinking.

This is just the sort of insight that students will come up with, given more time.  Where we can, let's slow down, and the let the thinking happen.  Curriculum can be a rush, yes, but sometimes that extra 5 minutes is the difference between getting somewhere with our thinking, and really getting to a more mathematical place with our thinking.

Jo Boaler and others have written eloquently about the problem of speed in working with students on their number fluency.  Slower but deeper thinkers can often be turned off of mathematics by our constant rush-to get from topic to topic, strand to strand, report card to report card.  Let's slow down and do the math!

Friday, October 10, 2014

Equal Signs, Balance, and Timbits (#tmwyk)

Callum had to start taking the bus to Kindergarten.  When I picked him up the first day, I decided to get him some Timbits as a reward.

(official looking Timbit image from Tim Horton's website)

A few minutes of driving passed, and I looked in the rearview, and suggested he save some for his brother.  To which, Callum replied, "there's only one left."

Nobody intends their child to eat 9 Timbits in one sitting, but it seemed like a good time to talk some math.

The conversation went something like this:
-if 9 are in your stomach, how many are left for Alec?  (one)
-if you started with 10, and 1 is left for Alec, how many did you eat? (9)

I wanted to get him to add "9+1=10".  I was working on the concept of "one more than nine, one less than ten."

Here is where things get interesting.  I use "plus" or "add" as much as possible, because they are specific operational words he is going to need really soon.  But based on recent readings, and #tmwyk conversations, I restrained myself from summarizing the math we were discussing as "nine plus one equals ten".

We are doing some work on Marian Small's Uncomplicating Algebra book, and the Ontario "Paying Attention to Algebraic Reasoning" monograph.  One common stumbling block is seeing an equal sign as merely an indication to "get the answer", or, even worse, what you press when you're done pushing buttons on a calculator.  More properly (and foundational to algebraic reasoning), an equal sign should be a statement of equivalence, or balance.

So I tried this:  "9 Timbits in your belly, and 1 more for Alec, are the same as the 10 we started with".  I wondered if "the same as" could be a good substitute for "equals" in talking math with young children.  I think small things like one word choice can have large effects on how young children learn math.  The precision of our math language really matters when we are building the conceptual foundation.

Consider this example (this mistake happens all the time in upper elementary):

4 + 5 = __ + 3

Those with less of an understanding of balance will right exactly this:

4 + 5= 9 + 3= 12

In grades 6, and 7, we see this exact error all the time.

Here is a powerful Kindergarten example showing how you can build the concept of balance with young children:

Any thoughts on the uses and misuses of equal signs?

Tuesday, September 16, 2014

Educreations in the Math Classroom to Capture Mathematical Thinking

This summer we were chatting about our favourite digital tools on Twitter and sharing some tools we would like to explore this upcoming year.  For my #Peel21st blog hop post, I wanted to talk about finally breaking through with Educreations.

The screen capture apps are well-known, as is the Educreations vs. Explain Everything debate (both have their pros and cons). I also know you can produce beautiful finished products with screen cap apps, but that's not what I want to talk about. I do think screen cap apps are a new text form all to themselves, but that is a subject for another post.

Pattern Clip

In the sample above, not much more happens than a short discussion of a pattern they built, but you can see and hear their thinking about patterns start to develop. Educreations work can be messy, and "in the moment ", and that's what we want in the math classroom. I have seen students who own the app use it as a sort of notebook, capturing thinking on the spot, even taking a photo of a textbook page and drawing all over it.

Here's another video.

Pattern #2

At this point you know we haven't developed long lists of criteria or a learning goal yet. We are still playing.

The inquiry cycle I prefer is this:
-play with the math
-look at what we did (in this case we watched about 12 pattern Educreations)
-bring out common misconceptions
-do mini-lessons as needed (on finding a rule from t-table, or graphing a pattern, e.g.)
-look at more and different types of patterns (not just blocks)
-develop criteria for the assessments we will use

Third very unique pattern

What I am saying here is that I like screen cap apps more for "in the middle" assessments than for summatives. You want conversations and observations all in one app? You got it! Evidence of mathematical thinking? Sure.  I think screen cap apps could be one of the most powerful assessment tools we have.

Puppet Pals by Debbie Axiak ​
IFTTT by Jason Richea
Notability by Phil Young

Wednesday, September 10, 2014

On a balanced math program, and knowing things, "all the way to the bottom"

You might know I got obssessed with Jordan Ellenberg's book, "How Not to Be Wrong."  It's not written for teachers, specifically, but it has lots of lessons and inspiration for us.

Here's one:
On Mathematical Knowing

I've spent lots of time thinking and wondering about why we spend so much time fighting about math. I jokingly talk about the #MathWars a lot, but in truth it's time to lay down our arms. One can stake out a position on the "back to basics" side, or the "discovery" side, but the truth is, and always will be, somewhere in the middle.

@PeelSchools, my employer, recognizes this-we have a balanced math instruction document now (as do several other boards).  There are many voices of moderation out there.  Practice needs to be balanced with problem-solving.  Number facts are the scaffolding upon which strong mathematical buildings are made, so they must be known.  Yes, students can "discover" a whole lot of math, but they usually need a lot of guidance to see what they found means.

One thing both "sides" agree on is- actually, forget that, there are no "sides". We ALL stand for student understanding, and being able to use math skills and concepts.  (One common debate is how and when we "know" a math  fact, versus how and when we "understand" a math fact- i'll leave that one to the researchers and cognitive scientists)

Perhaps the best strategy, in any given situation, to borrow from Mr. Ellenberg's quote, is the one that helps our students know the mathematical big idea or concept under study "all the way to the bottom."  If we are working with circles, that means the deep pleasure of finding pi using circles and string, AND developing, using, and applying the circumference formula.  Seeing how grade 3s can move beyond repeated adding as their schema of multiplication develops, AND help them start to know their facts with practice and games.  Watching junior students apply their sense of what proportionality means, and watch their toolbox full of strategies grow.

What we are not fighting about is the beauty and utility of mathematics-we all agree about that. Our methods and our means should help our students follow their thoughts about math deep down, all the way to the bottom.  Let them find the spark (with our guidance, of course)...      

Sunday, June 29, 2014

The Next Best Thing

I was thinking today, when I attempted to tackle some yard work with a 2 year old and a 3 year old in tow, that we don't always get to do exactly what we want to do. I can have a bunch of energy, and set out to pull every weed in the lawn, cut the lawn, trim every bush, sweep the patio, etc., etc., and I will pretty much always fall short.

You know that sick overwhelming feeling that comes when you see the weeds choking out the garden, and the patio covered in pine needles?  It's a tough one to overcome.  "I can't do this." "It's too much." "May as well just not start."

Then, at the end of the day, we ask ourselves, "have I done enough?" @tina_zita covers this question in her blog post, "June Comes the Same Time Every Year."  Doubt tends to creep in, and negativity. 

Teachers know this: we sometimes come in to work with to-do lists as long as our arms, a set of "must be dones", that don't get done.  Parents know this-dinner to cook, laundry to do, kids to put to bed.  Baseball players should know this, but they often swing for the fences when a base hit would do. 

There can be kind of an analysis paralysis associated with trying to get things done.  Too many things to do, too little time, nothing gets done.  But those who say just getting started is the biggest thing are probably right.  

I'm no expert with lifehacks for all situations, but today, I asked myself, "if you can't do everything you want to do, what's the next best thing?"

Cutting the lawn and pulling all the weeds became cutting the lawn close so the weeds didn't show.
Whacking the weeds down in the garden as much as I could took the place of weeding the whole garden. 
The patio got swept, but there is still a bit of loose debris. 

So try this thought exercise the next time you feel overwhelmed.  Take your perfect world set of goals for each day.  We are humans, and we dream big, so some of them may be a bit out of reach. For each one, figure out what the next best thing is, that you can live with.  

Mark 2 sets of essays instead of three.  Finish 3 items on your to-do list instead of 5, but do them really well.  Adjust your expectations, and be careful of your perfectionism.  

The lawn looks fine.  There's still weeds, but it's done. (For now...)

CC image by katerha.

Friday, June 20, 2014

"Put them in a line and count them": Comments from the Real/Fake World of Math Class

I used this problem with grade 6s yesterday:

I am not sure where this problem came from.  I do know that it stumped me for a while, although a decent number of students eventually got it, and the solution makes sense.

The first thing to note is that we are distinctly operating in the "fake world" here. No such club exists. There is no pressing need to do this particular math. There is no inquiry that can be done here. I defer, as always, to Dan Meyer on this topic.

 The problem is "word problemy", in the fact that it is asking for something simple- a single number of people, which it obscures through the design of its words.  That said, there are no dirty tricks here- it's a fraction problem, of the sort that uses a fraction of a missing whole, then a fraction of a slightly larger whole.

So why give this problem?  Here are some of our usual reasons:

1.  We are doing a fractions unit, and this problem can get us to think about fractions.
2.  To see what kind of thinking our students do on this problem.
3.  This problem is useful in its applications in the real world.
4. Because it's interesting, or beautiful.  These are our best reasons to do math, I think.

#1 didn't apply here.  #2 I am always interested in, although slightly less so when the pressure is off us in late June. As to #4, this problem did hold our interest for a while. It is a tricky one, and it engaged our desire to "puzzle it out".  There were some arguments interpreting the language of the problem, and trying and discarding some possible answers. Multiples of 4 and 7 were clearly involved, somehow.

As to #3, I will defer to my student's comments on the problem.

-you would just line the members of the club up in a line and count them. That's the solution. 

It's difficult to argue with that logic.

Wednesday, June 18, 2014

Reading, Writing, Coding, and "Absolutes" in Our Education System

I have taken a critical stance on the role of coding in schools.  Whenever I read an article with an underdeveloped thesis like "all kids must learn to code", I immediately ask "why"?  Often the articles are little more than a buzzy few paragraphs touting a great app. The "why" so often seems to be missing.

It doesn't help that these articles often come through aggregators like ASCD Smart Brief, Flipboard, or Zite with preposterous headlines like, "Is Coding as Important as Reading and Writing?"  Not only do the articles never make such grand claims, but also such absolutism tends to detract from the argument in the article.  (has anyone else noticed how headlines are little more than Buzzfeed style "clickbait" these days, even on reputable newspaper websites?)

I'm of the mindset that we never teach without the "why" in mind.  If we can't identify the critical skills, habits of mind, or plain old reasons for doing something, we probably shouldn't be doing it in the classroom. This post is an attempt to unravel the "why" of coding in the classroom.

My own experiences consist of a lone programming course in the Pascal language.  I am not sure that I finished my program for my project (a Minesweeper clone, that the teacher gave the perhaps politically incorrect name "Drunkard's Walk".) I do remember unraveling the mysteries of binary and other non base 10 number systems. That's a strong math connection right there. I would consider learning to write code for iOS, if I thought of a good app idea. Maybe that is a growth goal right there.

I've found lots of inspiring examples of code in the classroom, creative work in Scratch, for example, in primary.  My working theory, having seen but not worked in Scratch, is that it's an interesting and "new" form of visual storytelling.  I also think we should be listening to the words of people like athlete Chris Bosh, who talks about his formative experiences with code. I also usually tend to listen when the President of the United States stands up and asks all Americans to consider learning computer science.

This Mother Jones article is probably the best and most detailed read i've found on the topic.  The "why", if you accept the argument, and I do, is that we will all be the better for using the particular computational logic we can learn through becoming more computer literate.  Beginning from a "feat of imagination", and bringing creative, flexible and logical thinking leading to a task.  I think this is the true argument behind bringing coding to schools.

My colleague, @cashjim knows a lot more about this topic than I do, and here's what he has to say:

And further:

I am left thinking, like I often am, about the role of "non-negotiables" or "absolutes", particularly in the elementary grades.  There is a long checklist of curriculum expectations that every student must work on, at the same time as everyone else in their grade.  The breakdown of traditional subjects still holds-math is still math, geography still geography, history still past.  Everything stays in its little container (except when true inquiry learning takes root, and new branches grow).

So should coding be added to a list of non-negotiables for the 21st century learner?  What would it replace? Or should we be giving children as young as Kindergarten choices about things to learn, including coding? I'm of the mind that more choice is always better, and yes, I advocate for giving young children choices in what they learn.  So we wouldn't always do "coding for coding's sake", as Jim said: rather, give kids a problem to solve, and let problem-solving with code be one of their options.  Keep it about questions, and the constant process of inquiry (across all subjects), and let computational thinking be one of our solutions.

Friday, May 23, 2014

"Be Like Mike": Michael Jordan, Albert Einstein, and the Growth Mindset

In our math session yesterday at #7to10PDSBnet, we talked a lot about how having a growth mindset impacts on education.

As we were watching Eduardo Briceno's TED talk, I got to thinking about the premier athlete of my childhood, Michael Jordan.  I think Michael's early "failure" is well-documented (getting cut from his grade 9 team), but we probably don't often think about the growth, reinvention, and improvement that happened over  the course of his career.

My thinking is that it's every so easy to look at the Michael Jordans around us as freaks of nature, the lucky few who are so prodigiously talented and gifted that they make things look easy.  The truth is, Michael was probably the hardest worker around, and he never settled into a fixed mindset about himself as a basketball player.

Two examples to prove this point:
-mid career he became an average 3 point shooter. Not great, like Reggie Miller, or Ray Allen, but average.  But this did not happen without taking many many thousands of 3 pointers.  It comes out of a dedication and belief in improvement, and persistent work.
-at some point he added the ridiculous iconic turnaround jumpshot that became his main weapon.  Again, this doesn't happen without a belief in your own ability to improve, to get better, to grow.

You could probably argue that it's easier to have a growth mindset as an athlete-you are always working on and developing skills that will help you play the game better.  You could also argue that Michael had the benefit of height, agility, spatial awareness, and leaping ability.  There may be some truth to that- but his success was all in how he leveraged his gifts and grew over time.

We all have our gifts.  We just need to nurture them and develop them.  But what difference is there between saying "I can't shoot a 3 pointer", "I can't do Math", or "I can't cook"?  If we believe that we, and our students have brains that are constantly growing and developing, then the sky is the limit, (nearly) anything is possible. Such is the wonder of the human brain.

Another "anything is possible" kind of guy was Albert Einstein.  One should be careful about adding to all the maybe true/maybe false Einsteinisms on the Internet, but I came across a statistic about how many times Albert was wrong in his equations.  Simply put, he couldn't do the math.  He couldn't make things work out, over and over again.  What he considered his most important work, proving the existence of a force or energy that is pushing spacetime to expand, was a total failure.  It was proven much later, and the Nobel Prize went to someone else for that.  I have heard it called his greatest disappointment.  There is another statistic floating around, where a physicist claims around 20% of Einstein's work had mistakes of various degrees.

So perhaps if we truly believe in having a growth mindset, we can stop saying, "he's so smart", or "she's just not good at math".  Perhaps then "Be like Mike" wouldn't mean becoming a super athlete, but rather being the best you can be, and growing, learning, and changing over time, with persistent work.  Perhaps then being an "Einstein" might mean being someone who perseveres in the face of great difficulty, rather than "being a genius."

Friday, April 25, 2014

Gamification vs. Game-Based Learning

I should start this post with a bit of a mea culpa. In my CTV news clip about using Minecraft to show math work, I improperly referred to "gamification" of the classroom.  "Gamification could be the future of education," or something like that.

The distinction is probably easy to miss.  If one is playing a game in support of one's learning, perhaps you could be said to be gamifying your learning (if that's even a word).  "Gamification", though, refers to using game-like elements in the classroom, not necessarily inside a game environment.  "Game-based learning" refers to using a game in support of your learning.

This article by Jordan Shapiro is the most well-explained thing on this topic i've seen.He talks about how gamification systems are seen in places like coffee shops, with their loyalty programs.  Gamification, I think, broadly, could be said to be about modifying human behaviour to better reach a goal (free coffee, perhaps, or better learning outcomes in Math class).

I loved his examples of video games we learn from.  I agree:  students learn from every game they play, whether it's the new "2048" app craze (powers of 2), Angry Birds (physics/geometry), or Call of Duty (single-minded concentration and pursuit of goals, teamwork, reading and writing skills when playing online with others, communication).

It's interesting how viewpoints have changed on the video game issue.  I remember even 6-7 years back debating their merits with students.  It would seem they have been allowed into mainstream culture (and educational culture) in a way that wouldn't have seemed possible even a few years ago.

Here is where I write a letter to my 13 year old self:

     Dear Matthew,

     20+ years from now, you won't have to hide your comic books, and video games will be everywhere.          Have faith and be patient.  Let's hope Zack Snyder doesn't ruin Bruce Wayne like he ruined Kal El!  PS:      you have two children and you don't have time to play games, but all libraries have lots and lots of comics      for you to check out.

    Older, more bearded you.

But seriously though, the games I did play, I often played in a completist way- finishing every single goal in "Super Mario Galaxy" for example, or playing all the bonus content in "Resident Evil 4".  As years went by, I found I would focus only on the main goals, and not the side goals.  In Mario terms, completing the course, from start to finish, without focusing on extra start, or any extra content.

Therein lies my problem with gamification. If we are truly focused on big ideas and learning goals, side quests (for example to get "badges") might get in our way.  I also believe in naturalizing the classroom environment as much as possible- so for me, gamification systems would only get in the way of conversations and interactions.  Lastly, and my main concern perhaps, is that focusing on gamification systems might undermine students' intrinsic motivation.  Put it to you this way:  if I don't need a coffee, or even want one, and I notice I need one more sticker on my McDonald's card to get a free one, will I go?  Further, a quick search on this subject reveals many corporations are looking at gamification strategies to modify their consumers' behaviour, which doesn't bode well for it's future in education.

Game-based learning suits me a lot more.  Minecraft, for example, has been absolutely amazing as a tool in Math class.  But what's worth noting there is the differentiation:  students CHOOSE to become immersed in a  game environment, and only IF they can meet their learning goals with regard to the mathematical concept or big idea under consideration.  The other big thing here is that there are NO goals when one opens a world in Minecraft.  There are no badges, achievements, or levels.  Perhaps Minecraft is the least gamified of all games; even earlier sandboxes like Grand Theft Auto had missions that you could choose to complete (and most players probably did).

Game-based learning is no simple panacea.  Differentiation is key. Many games are too closed ended to be useful in the classroom beyond one single set purpose (and there is nothing wrong with that, in those instances).  Let students make choices about how they learn, and if it includes video games, so be it.  My 13 year old self, and perhaps some 13 year olds in your class, will thank you.

Wednesday, April 23, 2014

The Ingredients for Math Class Are Everywhere

Remember those teachers who could turn anything into a teachable moment? Perhaps something happened on the playground, or the world outside the classroom, and they turned it into a lesson.

If you can think of someone, were you thinking of your math teacher? I started thinking about teachable moments in math when I was shown this on the playground.

The student said: "I have a dollar."  He had taken the original $2 coin and knocked the centre out with a hammer and screwdriver. 

The picture I took after the conversation was enough of a provocation on its own.  All that was needed was the question:  Is it worth a dollar?  If not, how much? The most likely relevant technique for coming up with an answer (area of a circle), is taught in the same grade level.  

Here is the task in a tweet, with a picture made with PicCollage and Skitch:

I've been thinking a lot lately about why we have always felt the need to rely on outside sources for our math questions, problems, and tasks.  Once you start to notice it, math is everywhere.  This is a mindset- much like the English or Social Studies teacher has when she finds articles, texts, or news stories that relate to her tasks, lessons and units all around her.  Or how about the Science teacher, who is always using the latest developments, innovations, and news as the raw materials for her lessons?

We can find authentic, meaningful and completely contextualized materials for math lessons all around us.  In planning for some Ministry of education work (a webcast on proportional reasoning), I spent a few weeks ruminating on what the tasks should be.  And then I went to the movies.  This visual sparked 3 days of work for 3 different grade levels:

In this case I was actively hunting for materials for rates problems, but my math antennae are up these days, and they are getting good at finding the raw materials for math.  Math is everywhere, and all we need is our tools like our camera rolls to record them.  

The value of the toonie ring could be $1.69, but that's not accounting for the differences in the metals, or their weights.

Friday, April 11, 2014

The Road to #GeniusHour Math

The road to Genius Hour Math began with a bunch of sticky notes stuck on a blackboard.

Inspired by the concepts of 20% time, passion projects, and the great Genius Hour work happening in schools all over the world, I set out to explore what this would like for those of us who don't have our own class (and do math on a rotary schedule).

My hypothesis was this:  our math curriculum (an incredibly balanced and strong one here in Ontario) is short on things like the history of math, or the math that makes the world work, or bigger topics that are a little outside the traditional elementary math box, like infinity or dividing by zero.

I wanted my students to explore some math of their choice, so I had to see what topics they were interested in first.

As I am #poweredbykids, I started by watching their brainstorming.  My only question was this:  "what's something you wonder about that can be answered with math?" I exaggerate little when I say that I could plan my entire math program by just letting them ask questions.  Once the inquiry math mindset is open to them, all the old fears and hatred about math fall away.  They become questioners, and they start to make meaningful and important connections to the math in the world around them.

The board (now called the Wonderwall) filled with stickys with questions like:
-was math discovered or created?
-I wonder what the last number is?
-is there a pattern that hasn't been thought of yet?

One student wondered how much of a certain substance Toronto's notorious may had consumed, but that question, while mathematical, could not be answered.

These were clearly the raw materials for something really good, but more work needed to be done.  We took the odd question down to explore it, but mostly the board just sat like that, for a good two months. One day, reading the questions, it occurred to me what had happened with the brainstorming.

All the questions fell into one of three categories:  Googleable, estimateable, or needing more exploration.

Once this realization happened, the hour of our greatest genius was upon us. We watched this astoundingly beautiful video on the beauty of math.  I used Numberphile's video on cryptography to show how complicated mathematical concepts could be distilled into a carefully crafted explanation.  I showed Kid President's Pep Talk.  I explored dividing by zero with them to spark their minds.

As inquiry learning begins with questions, I gave them a simple organizer asking for a big question, and two sub questions that could be explored on the topic.

I discussed these with them, and offered feedback which would help focus the projects.  We developed some simple success criteria for what the project could look like.  The evaluation could not be pegged to a specific strand, as it usually is (although everything could broadly be said to be about number), so we mainly looked at evidence of mathematical thinking, and how it was communicated.  The Ontario curriculum front matter, particularly the achievement chart, is the math teacher's best friend.

From there, they ran with it.  The quality of a lot of the projects exceeded expectations.  Topics ranged from things like types of infinity, to music in math, to statistical analysis of a number of sporting themes, like determining who will win the NBA championship.  Activating the math of sports could be the topic for a whole other post.  Students do amazing work when they are shown how much math is in their favourite sports.

One of the most memorable included a live demonstration of poker probability, in which the dealt hands (under the document camera), provided proof of the odds that were posted on a chart of the wall beside.  Another sought to prove whether Santa Claus could make all his deliveries in a single night.  

The only slight challenge I would say is helping your students make more research-based topics come alive (like infinity).  Some projects were more research-based, and less based on mathematical technique, while the best of them had students actively developing their own mathematical technique to explore their topic.

I would encourage anyone to start on the road to #GeniusHour math.  It's worth it.  Too often (and for too long), we have let math ferment in its little silo.  Strands and units create artificial boundaries that prevent students from making connections to their own lives.  Math is seen as something that comes from teachers, in specific class periods, at specific times of day.  Finally, we are too often the questioners in our math classrooms.  Let your students be the questioners, and see what happens!

Tuesday, March 18, 2014

6Cs of 21st Century Learning Project: Character Education and Citizenship

In which, @jasonrichea, @MatthewOldridge, @DebbieAxiak and @tina_zita finally finish their writing project on the 6 Cs of 21st Century Learning.  


The two newbies to the group: Character education and citizenship. Some will tell you there are only 4Cs of 21st Century Learning. Character education and citizenship won’t be found on many lists. I must admit that when I read them listed in Fullan’s From Great to Excellent report I was a bit surprised. They seem like a given in education, something we have championed for a long time. Maybe that is one of the reasons this post has been the hardest of the group. As i’ve been pondering this blog post I realized that perhaps they were included for that exact reason, so that we could take a closer look at what they mean in our digital age. The other Cs aren’t much without character education and citizenship.

When thinking about the two terms in a digital era the following comes to mind.

Character Education in the 21st Century means understanding that the digital and physical world may seem far apart but really are one in the same; it is helping students portray their best self both in face to face situations as well as during online interactions; it is the kid that lends a helping hand to a friend as well as the kid that will refuse to pass on that photo coming around.

Citizenship in the 21st Century means understanding the power that we each possess to make a difference within our community, our province, our country and beyond. It is understanding how it is not just a right but also a responsibility to be aware. Digital citizenship is definitely a part but not the only piece that is important to discuss in today’s classrooms.

Collaboration, Communication, Critical Thinking and Creativity are wonderful skills but without character education and citizenship they aren’t really given any agency. Through out the post I have been thinking of Kid President and his positive message calling us to action. That would be my hope for the students in our classrooms today.

Character education = being and sharing our best.
Citizenship is taking action.

Character Education

@Debbie Axiak

Character Education – Learning to do the right thing

Martin Luther King Jr said “Intelligence plus character - that is the goal of true education”. Schools play a role in the development of the whole child - teachers don’t just teach subjects, we teach people. Parents, family and other community members, and schools play a huge role in the development of character by instilling the values of what it means to be a person of good character. Being kind and caring to yourself and others, telling the truth, accepting responsibility for your actions, doing what you say you will do, respecting differences and cooperating with others are all part of what makes a person of good character  – at home, at school or out in the community.

Wouldn’t it be great if we were all good people ALL the time? Well, we aren’t perfect - we are human and we make mistakes. Children, teens and adults are constantly learning as we encounter new situations. Being kind and caring (or honest, respectful, inclusive, etc.) feels a little different when you are a three-year-old at home with a loving family than it does when your ball gets taken on the playground, at your first teenage party with no adult supervision, or at a U.S. Walmart on Boxing Day.

People of good character are the ones we want to be friends with, the people that we would hire, the people that we respect. When we are surrounded by good role models and are held to high expectations throughout our formative years, we can internalize what it means to be a person of good character and do the right thing, no matter what situation we find ourselves in. Sadly, some of us come from homes where selfishness, racism, homophobia, lying, cheating and/or stealing are the norm and school becomes the venue where we learn about the importance of becoming, and being, a person of good character.

In every case though, becoming a person of good character takes time to develop.  We must experience many different situations and people, we need to have good role models whose actions and words demonstrate the power of character, and we need the time to prove ourselves as a person who does the right thing no matter what the situation is.

**Added after reading @jrichea’s response - Jason said that in high school, Character is not something that is often specifically discussed. This is different in the K-5 and 6-8 schools where Character Education is specifically discussed by teachers through a Character Committee. Each month a different character attribute is highlighted through Character Assemblies, through announcements, by celebrating student actions in relation to the Peel’s Character attributes and through in-class activities (often integrated into Language Arts by reading, discussing and writing about the Character attributes seen in fictional and real characters). The attributes are integrated daily in K-5 schools as students learn the importance of being honest, caring, respectful, etc. in a school setting, at recess, etc.

I guess this difference is expected. Younger students require more explicit instruction during the early school years as they learn to navigate in the world away from home and become more responsible and mature.


This is a story from the early days of Facebook, the dark days before we started to bring digital citizenship fully into the light, and schools were equipped only with the most scary and dire media stories about this new menace to youth.

I was home, cooking, early in July.  This was for about the third date with my future wife.  I received a phone call from a parent of a girl who had been in my class.  She had been targeted with the sort of “you should die” chat bullying that happens these days in digital mediums (and happened in other ways, long before that).

That was upsetting enough, but it also turned out that a bunch of us (staff at the school) had been tagged in a Facebook photo in profane and unflattering ways.  I was unable to see the photo, due to privacy settings, and that in itself was a helpless and horrible feeling.  No calls to board or union helped- we were in uncharted territory here. Nothing could be done.

In this case though, all the parents involved have great parents who helped work out a difficult issue.  Atonement was made, but not to me, and I had to accept that it never would be.  I had it on good authority that my image was taken down, though.  The final authority on the character of the bully was the girl’s parents (as it should be). They were loving mother and father, and her first and most important teachers.  

I am sure she has grown into an excellent person.  I personally wouldn’t be who I am without making a whole mess of mistakes as a younger man.  The notion that character is fixed and immutable belongs to comic books and “Criminal Minds”.  We become who we are, slowly, steadily, as the sum of all our experiences (and those who guide us) take us down the road through childhood, to adulthood, and so we continue until the end of our lives.   

Character is becoming.

Jason Richea @jrichea

I find character is not something often discussed among educators. Not specifically anyways. We often share our opinions of students and how they behave in the classroom, but never really attach the label ‘Character’ to such discussions. We look at how they collaborate, communicate, work in teams, show initiative, determination, responsibility, maintain focus, organize themselves, and work independently as well; but never do we judge character specifically.

However, in other arenas character definitely becomes a talking point.I have had the pleasure of coaching many student-athletes, and this is where I really the character in students come out. The classroom can allow for such exposure, but I find the playing field is where true character can really shine. Many of my players have demonstrated sportsmanship, respect, admiration, teamwork, work ethic, determination, commitment, sacrifice, and what it means to “leave it on the field”. We often coach players on character, or sportsmanship in this case, and are explicit in its teachers. We reward players for demonstrating such character, and often find ourselves in regular discussions among coaches as to whether to take a skilled player over a player who displays a tonne of character? The answer is never clear, and context must be given, but more often than not, the choice is character.

So how do we instill character in our students in the same way? What attributes should we develop? I think it comes from leadership really. If we can develop our students to become leaders in their schools, then we are instilling in them many of the qualities that we look for in someone who has a tonne of character. They will lead by example, respect & listen to the opinions of others, look to harness the skill sets of their team, and take responsibility for all actions. In essence, true leaders, have true character.

Character = Leading by example



Citizenship – Contributing to a better world

When I was a girl, citizenship meant helping others in the classroom, school, neighborhood and religious community. Once a year, on Halloween, it also meant carrying a UNICEF box and collecting pennies for nameless, faceless ‘starving’ people in faraway lands (these were the same people I was supposed to feel guilty about every time I didn’t finish my dinner).  News came from the radio or television a few times a day – but I didn’t listen or watch the news as a child. As a lower-middle-class WASP girl growing up in the First World in the 1970s, war, starvation, environmental disasters, and human rights issues were abstract ideas that I had very little exposure to between episodes of Get Smart and a great game of hide and seek.

Today, news updates are constant, video footage and photos give equity and social justice issues names and faces, and people from faraway lands live next door. Students are aware of local and global issues and they want to talk about them and do something about them. I’ve had middle school students come to school wanting to discuss the dire conditions in Attiwapiskat schools, want to fight for Shannen’s Dream, stop Global Warming, or talk about Joseph Kony and his child soldiers. War, starvation, environmental concerns and human rights issues are not abstract ideas for today’s students.

As an educator, I should go beyond talking about world issues with my students and help them make a difference but I struggle with this. Luckily I’m surrounded by phenomenal students and teachers in my new school and the Outreach Club’s dedication to raising awareness and money for a variety of issues is helping me learn how to go beyond words and to take action.

Today, citizenship isn’t just about contributing to the local community; it is about making the world a better place.

Jason Richea @jrichea

I am extremely fortunate to be a teacher of Canadian geography. Geography allows you to see all of the interconnections in this world, and learn about the problems plaguing this planet; while at the same time develop solutions to such problems. Contrary to popular belief, it is not about colouring maps, and labelling the provinces & territories. Not only do I teach geography, but I find myself a student of this subject as well, due to  the various events happening around the world, every single day. One of the great aspects of this, is that I am continuously learning about Canada’s role in this world, and what it truly means to be Canadian.

So what does this have to do with citizenship? Well quite simply, Canada is an amazing country whose actions (for the most part) attempt to improve this planet. These actions really are a reflection of the citizens of this country, and led by individuals and groups of Canadians. To wit, I always find myself astounded by the way in which ordinary Canadian citizens attempt to improve the lives of millions around this world. Canadians regularly reach out to victims of catastrophes, build villages, start non-profit organizations, share their knowledge, skill, and expertise, and all because they want to make the world a better place.

The best part of all of this, is that these actions are not limited to the adult citizens of this country. Youth are engaging like never before, and leading the charge to change the world. The awareness they have for world events is astonishing, and I am in awe of their passion. Our students regularly display citizenship, just through the compassion they show for those less fortunate. They hold food drives, bake sales, runs for the cure, and a variety of other events, all organized by themselves in a lot of cases, because they refuse to wait on others to find solutions. This compassion and drive is what really sets Canadians, and our students, apart from others.

Citizenship = A Passion to Improve our World


I am Canadian. I stand on guard, or at least, I would, if asked. I used to have all our Prime Ministers memorized. I belong to God and Nancy and Callum and Alec.  I am Mississaugan, but I came from Guelph General Hospital, and my mother and father, and their mothers and fathers before them.  I am #PeelProud, and I belong to the #Peel21st.  I left a small piece of myself at a shrine near the Pacific Ocean in Japan.  I found another piece at the corners of Keele and Dundas Street in Toronto, although I am not a citizen of Toronto.

As @neiltyson would say, I am from: EARTH – SOLAR SYSTEM  – MILKY WAY GALAXY – LOCAL GROUP – VIRGO SUPER CLUSTER – OBSERVABLE UNIVERSE. Although based on recent findings, we may have observed all the way back to the Big Bang, and that's not to speak of possible multiverses, and earlier occurences of this universe, and the possibility that I am as I am, typing these same words, in many different times/places...

I belong to all these things.  Citizenship is belonging.

Thursday, January 23, 2014

Add Calculus To Your Workflow

This weekend past was the one where everyone was working on their report cards.  I know this because:
1) I was working on report cards, and
2) my entire Facebook feed seemed to be updates on where everyone was at on their report cards.

I got to thinking about teacher workflow.  Our work is usually more incremental, period by period, day by day, unit by unit, that larger projects such as report cards tend to throw us off.  Stress levels go through the roof.  I am not immune to this, myself, but I have reflected on when I feel the best, and why.

In reading Ian Stewart's In Pursuit of the Unknown: 17 Equations That Changed The World, I just read the chapter on Newton (and Leibniz, almost simultaneously) discovering Calculus.

Calculus is often defined as the math of change, and one thing it allows us to do is calculate rates of change. Another is integrating to find the area under a curve.  We might imagine the area under our curve as representing all the work that needs to be done on a given project.  Now imagine this example for your typical teacher during the report card period:

On our work vs. time graph, if we were to take any unique point on the line, we might not feel like progress is being made.  But if we divide up our workflow into enough unique points (for example, making a day by day plan for the entire reporting period), then it may seem easier.  As long as our rate of change is positive, we should feel as if we're making progress.  As we get to the end, momentum applies, and we may feel like we're accelerating our rate of work, simply because there is less to do, and as we get toward the end of the project, things will kind of taper off, and we get to relax.

This may seem obvious, but looking at how much work we have to do all at once is a sure mood and morale killer.  We may just give up, and not start "climbing the curve" until it's too close to the deadline, and stress and panic set in.  Instead, check where you are at any given point in time.  Set manageable time goals, and meet them.  Planning day by day for a week ahead may give you enough data slices to feel like you're making progress.

Wednesday, January 15, 2014

Assessment Should Not Be An "Event"

In working on IEPs this week, I found that the categories for "Teaching Strategies" and "Assessment Methods" were blurring in my head. If I had to pinpoint why, it's probably because I see them as so closely connected as to be near indistinguishable.  Things that would fit in one category, would also fit in the other, in many cases.  It is very clear that "conversations", for example, fit into both categories, especially when you consider that conversation is one of the 3 pillars of assessment from Ontario's Growing Success.

We've heard a lot about assessment for, of, and as learning.  We know the importance of diagnostic and formative assessments, and we still use big summative tasks that connect to big ideas and units of work.  But here's the thing:  assessment IS learning.  Provided I am not going to fall back to a lecture style, in which I talk too much, and completely bore everyone, including myself, then the daily life of the classroom is constant discussion, observation, negotiation, sharing of ideas, and the continued blurring of lines between teacher and learner.

I say "blurred lines", in this case, because we all learn together, and if I didn't see myself as a co-learner, then what am I doing this job for?  I learn and am taught things everyday by my students.  I may hold a higher level of math knowledge that I need to share, but that's just the job of a coach.  (I did get schooled on the value of 0/0 the other day, but I digress)

It may also be that I have come to consider "evaluation" as the least of my three roles (the other two being talker, and observer).  We do big projects sometimes, and we are proud of them.  But they are but the final step on a long journey. Percentage wise, we spend much less time working on them as well.

There's one more thing:  evaluation, necessary though it is, often occurs on artificial timelines.  I NEED to finish Geometry now, therefore the evaluation must happen, or I NEED to get to the next unit in Geography, or report cards are due next week, is my Drama unit done?

Real life doesn't work that way. Nor should it.  Learning is our main work.  We talk and learn and explore, and I coach, but mostly I learn too. In #cdnedchat the other day, I responded to a tweet from @DanaAriss, as you can see below.

How far along have we come in changing our thinking about assessment and evaluation?  What do you think?