Roll Up The Win To Win, Odds, Luck, and the Law of Large Numbers

Pre and post ceremonial losing cup stomp. #timhortons #RollUpTheRim #odds #probability pic.twitter.com/qLz8fd9Wp4
— Matthew Oldridge (@MatthewOldridge) February 8, 2015

A Warm Cup Full of Hope?

"Roll Up the Rim to Win" season is a yearly warm cup full of hope in the otherwise dreary month of February. There is a car at stake, after all (one for the whole country), and the winner could be me.

Here is how the narrative of luck plays out in our heads:  I realized after the fact that I threw out a cup in Target (speaking of dreary Februarys).  Briefly, it crossed my mind,  "what if it was a big prize?"
We might feel the same way if we threw out a lottery ticket.

Here are some reasons why I shouldn't worry about it:

-The stated odds (from http://www.rolluptherimtowin.com/en/in-restaurant) are 1/6 to win any prize. I only had about a 17% chance of winning *any* prize. I can live with forgetting about that cup.

-there are 3 cars to be won in Ontario.  There are some 143,000,000 cups being sold in Ontario.  See here: Restaurant Rules. This is statistically zero. Actually, do some lotteries have better odds?

-there are 94 t.v.s to be won in Ontario.  Again, statistical zero.

-there ARE 23,000 $100 gift cards at stake.  I calculated a 0.16% chance of winning one. Again, I can live with not rolling that cup

-that means the 1/6 odds basically apply to food prizes. There is a 17% chance I missed out on a doughnut.

-the picture above is from the first cup i've had this year.  I didn't roll the second, because I threw it out. Expecting a streak of 5 losses in a row is how you might need to think about it.  Then again, you can sometimes roll a "six" in a dice game, when you need it...

The Law of Large Numbers, and Our Perception of Luck

I will let you read the Wikipedia article on the Law of Large Numbers. The  "house always wins" rule, you might call it.  One of the most brilliant ways to teach it is to use a coin flip simulator, combined with the kids actually flipping their own coins a number of times.  See the Virtual Manipulative here. They watch the sim go to very close to 50/50 over just a few thousand trials, and understand that the odds must come true over time.  How do you think casinos stay in business?  Because they *know* how much they will lose and gain, as an average, on every single bet.

I used to have kids record their flips like this:

e.g. HTHHHTTHHTTTHHTTT...

There is an amazing trick you can pull on them. Have them record one real trial of 50, and one fake trial.  You will be able to guess which one is fake, every single time!  Why?  Simple-our brains perceive 50/50 as truly random, so kids will make their fake trial *too* random!

Real trials of 50 will usually have a streak like:

HHHH or TTTTT.

Look for those.

How does this apply to Roll Up the Rim?

Let's say you have won 3 doughnuts in a row.  You feel lucky right?  You might even feel like you should go out and buy a lottery ticket.  Wrong- after you have had a streak of good luck is the *worst* time to go out and buy a lottery ticket.

Here is your streak:

WWW

But what cups did you have before, and how many after?

Is your true distribution something like this?

LLLLLWWWLLLLLLLWLL...

4/18, or just ahead of the odds. I suppose that would be a bit lucky.  But there's a reason for the saying "quit while you're ahead..."

We're talking about math here, folks, so this even has a name:  Poisson clumping.  Random events occur in bursts.

Now that we've talked a bit about Roll Up the Rim odds, here are your choices:
-ignore the tabs, because there is a 5/6 chance every single time that you have lost
-keep dreaming of that new car!
-just enjoy your coffee and have fun

Breakthroughs, "A-has", and "Finding the Door"

I highly recommend this article on Yitang "Tom" Zhang's breakthrough on the Twin Primes conjecture. It's readable, inspiring, and full of lessons for elementary and secondary teachers of mathematics.  Don't worry about understanding the math (although it does explain it pretty nicely), just read Tom's story, from unemployed math PhD helping out at Subway, to becoming a professor, to his big breakthrough.

First, our students have these kinds of breakthroughs each and every day.  They don't have to be Archimedes style "eureka" and "jump from the bathtub" breakthroughs.  They aren't Einstein imagining himself on a beam of light, and conceptualizing relativity.  But they are breakthroughs, nonetheless.  Rather than bolts from the blue, these are often subtle and sudden realizations that they are closer to solving a problem than before.  You know, the "a-ha" moments.

These are the moments when things become clear, when they can see their way through a problem. They can begin to articulate a solution through the complex interaction of math processes, content and background knowledge, and classroom culture that we call problem solving.

There is a lovely description in the article of Mr. Zhang walking around and thinking about the problem, until one day, he "found a door".  He realized what tool he needed to solve the problem, and how he would do it.  The unbelievably complex math he was working on began with selecting tools he would use, and finding a way to represent the problem.  Sound familiar?  They should- they're two of our math processes.  If the math processes are the "actions of doing" math, doesn't it make sense that a professional mathematician would use the same processes as our young learners?  I think so- practice, experience, and knowledge background being the main difference.

The growth mindset math learner might say, "I can't solve this problem...yet."  We might change that to "I haven't found the right tool...yet."  Or:  "I don't know how to represent this situation with math...yet."

We could also say, "I haven't found the door...yet."  After you find it, you can walk right through, after all.

10 Good Things

I was challenged by Brian Aspinall to do a #10goodthings blog post, so here it is.

So many good things happened in 2014- reflecting back makes me very happy.   Through good fortune a lot of things fell into place.  These are my 10 good things from 2014.

1)  Minecraft and Math happened.  See video clip about the work of our classrom here.  The real hard work of using Minecraft as a consistent classroom tool is being done by people like @GumbyBlockhead and @zbpipe, and all the schools around the world on the MinecraftEDU servers.  

For me, this was less about technology than about an opportunity for differentiation and student voice.  What will happen if I let them try the math task (or even a textbook question!) in the Minecraft environment.  The results far surpassed anything I would have thought possible.  I kept asking:  "are you sure this will work in Minecraft?", and kept having my skepticism proven wrong each time. 

My blog post about the journey is here. 

2)  The journey into inquiry math.  Pictured is a student's projections using their mathematical model for how many points these players will score in their careers.  We use "wonder as the fuel" in our #GeniusHour math, and saw how questions spark more questions, which spark more learning.  Letting students loose to explore a math topic of their choice was amazing.  Their work exceeded all expectations!  My account of #GeniusHourMath  is here.  

3)  Starting #PeelMathChat with @raspberryberet. Here is a Storify of our very first one.  It is always inspiring to see educators talking math in the evenings, when they could be binging on "Fresh Prince" on Netflix, or knitting a scarf.  But seriously- our PLN is strong, and this is one way we share and collaborate.  

4)  Social time on Twitter (and IRL) with friends like Carla Pereira, Jonathan So, Tina Zita, Helen Chapman, Debbie Axiak, Melanie Essex, Jay Richea, Donald Campbell, Aviva Dunsinger.  Because, you know, it's not all work, all the time.  #Peelnomnomchat and other fun chats often bring a certain needed levity to life.  Now let's all help #Peelsockchallenge take off as a hashtag!

@Carla_Pereira2 these #noms are back! pic.twitter.com/j81eNIsMKm
— Matthew Oldridge (@MatthewOldridge) January 20, 2015

5)  Presenting about Math and apps at #TLDWpeel in the summer (which led to a further 2 part workshop with @tina_zita, and my #BIT14 presentation).  Sitting on the expert gaming panel about Minecraft was amazing too. 

6)  Watching @MathletePearse at #BIT14, and having to sit in the hallway because it was so packed.  Doing my session right after, 3 doors down, and having the room totally packed.  

Also at #BIT14, the power of the people in creating change in education:  the informal yet amazing discussion with @MrSoClassroom and @MrAspinall at #BIT14 on assessment and math (in turn, sparked by a Twitter conversation) spoke strongly to the power of informal networks of information sharing and professional collaboration. I also turned around and @avivaloca was right behind me!

7) Harnessing the power of the #MTBoS to improve my teaching and learning.  When you have a worldwide network of people like Dan Meyer and many others sharing daily, you are never short of ideas.  

8)  Being the host classroom for the learnteachlead.ca project "Loving the Math, Living the Math." This was the most inspiring 3 days of teaching I have ever had.  I'm not even sure what to say  about it.  I've been tweeted from across the province ("i'm watching your video") and even had someone say, "i've watched all your videos."  I can't even put into words how many things I learned about math teaching and learning through this project (or rather, it would be a very very long essay, and a topic for another day.)

It's not me though- it's our amazing Peel students who are willing to grow their brains in math, take an inquiry stance every day, and #EngageMath with us. Together we are strong. Believe in the power of kids, let them be kids, let them talk, and engage their natural curiosity about math and willingness to learn, and they will do great things!

9)  Being hired as an Effective Math Resource Teacher in @PeelSchools. This was a goal i'd had for over 5 years, and included a handful of unsuccessful interviews.  For my personal journey, I think of the Japanese proverb (and personal mindset motto), "fall seven times, stand up eight."

It is the greatest privilege to be able to engage math every single day with our amazing school board-students, teachers, principals, superintendents, everyone.  

10) Trying to live as ethically as possible, and being the best father, husband, and human being I can.  

 Is your hourglass 25% or 75% full in 2015? Do we focus on what we have, or what's missing? 

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.

Math process of the day. How do we create representations of something so abstract as mathematics? pic.twitter.com/tUx71Rvw8r
— Matthew Oldridge (@MatthewOldridge) November 28, 2014

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!

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:

RT“@PlayAndWonder: #engagemath talk about the conditions needed to balance the scale. @peelfdkmath pic.twitter.com/PfKYt64X7q” @MatthewOldridge
— Peel FDK Math (@peelfdkmath) October 9, 2014

Any thoughts on the uses and misuses of equal signs?

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.

Code.or by Erica Armstronghttp://msarmstrong.ca/2014/09/my-favourite-tech-tool-code-org/

Puppet Pals by Debbie Axiak http://debbieaxiak.blogspot.ca/ ​

Scratch Jr. by James Cash http://makelearn.org/2014/09/16/scratchjr-peel21st-blog-hop/

IFTTT by Jason Richea

beyondangrybirds.blogspot.ca

Photo Editor by Aviary by Tina Zita http://misszita.wordpress.com/2014/09/17/peel21st-blog-hop-my-tool-for-this-year-is-photo-editor-by-aviary/

Pocket by Greg Pearson http://leaderinadigitalworld.blogspot.ca/2014/09/cant-find-somethingyou-shouldve-put-it.html

Notability by Phil Young http://wp.me/p3RGo2-1Jb​

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)...