What Math Lessons Can We Learn from the Game of Nim?

Have you played the game of Nim before? Do you know what lessons we can pull from the game? Watch me play the game with two of my daughters Jules and Lucie.

You can see right when the game gets to 4 left that each girl knows they lost! You can see it on their faces and even Lucie explains it to us by giving us all the options. Jules even wonders out loud “How did you do that” She knows there’s some trick here.

So, let’s see how I’m the World Champion at this game.

This game is a variation on the game of Nim

“Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap. The goal of the game is to avoid being the player who must remove the last object.” 

I also play this game with my students. The interesting thing about this game is that there is a winning strategy. And in our variation of the game if the player to go FIRST knows the winning strategy then they are guaranteed to win. So even though it looks like I’m the World Champion it just comes down to math.

Let’s explain,

Jules and Lucie know they have lost the game when they are left with 4 to choose. So as a player if you can leave 4 objects for your opponent to choose from you have won the game!! So now the game becomes who can leave 4 for their opponent to choose from. How can you always get to a position where you leave 4 for your opponent? Think about it for a moment. What number should I leave my opponent to choose from so that no matter what they do I can then leave them with 4 to choose from?

Right..l Should leave them with 8 to choose from! And then where’s my next winning position? 12 then 16 then 20. Multiples of 4!

So right from the start since there was 21 objects in the pile I can get to 20 on my first move by going first! And win the game every time.

In my class I usually put some cash down on the table to enhance the experience. “Anyone who beats me at the game will get the cash!”

Every time I play this game I’m reminded of my math education as a student. You see, in the game of Nim if you know the winning strategy you win every time. You know the path to follow. You see how it works. If you don’t know the strategy you are playing the game almost as if you are blind. You’re not sure how your choices will affect the final moves near the end. You are hoping the moves will pay off down the line.

As a student most of my math educational experience was like the experience of the player in the game of Nim that doesn’t know the strategy. I followed the teacher (who does know the “strategy”) blindly. I wasn’t sure of how my “moves” would pay off in the end. I just followed the rules hoping for good outcome.

I was such a good rule follower that sometimes it awarded me some success. In the fourth grade I remember earning one of those big puffy, stick off the page stickers for being a master multiplier. Yay go me!!


But when it came to being pushed to show my understanding the wheels fell off. Here is a 4th grade test on multiplying (when I look it over now it looks like I must have fixed this up after getting it back). Math for me was like a series of tricks that I could memorize and then try to perform.


Thinking back to playing the game with my daughters you can hear Jules, the first girl in the video ask right at the end “How did you do that?” She was thinking this is all a trick! The game was like a magic trick. How many of our students see their math education as a series of tricks? Lots of them I bet.

We don’t want kids thinking math is just a series of tricks to memorize. If they do think the math they are learning is a trick then it’s our duty to uncover the trick. Show them how it works. Like in the game of Nim students should know why the first player has the winning strategy.

This is what I want from my math lessons. Let’s continue to fuel sense making in our students instead of showing them just tricks. So, in your next class play the game of Nim with them. Blow their socks off and win 3 times in a row…..but don’t leave it as a trick. Uncover the math and strategy behind it together!

A great read is Nix The Tricks by Tina Cardone https://nixthetricks.com/

You can read more about how to fuel sense making for students from Kyle Pearce as he describes a task around Donuts.  https://tapintoteenminds.com/3act-math/donut-delight/

and also here from me showing a task about the defrost function on my microwave http://mrorr-isageek.com/fuel-sense-making-black-box-defrost/

Why Consistency Is More Important Than Intensity: Culture in the Math Classroom

Ok so you tried using a problem based lesson like a 3-Act Math Task or maybe you had students solve a task in groups with your brand new whiteboards and it …….flopped! Yep you’re worst nightmare was there with you in that classroom! A failed lesson! You were sure it was going to be a success. You heard that that activity was a great one but for you it just didn’t work.

Don’t worry. It wasn’t you. It’s normal. This comes down to an example of Intensity versus Consistency. I first heard Simon Sinek (an inspiring speaker and author on business and leadership) talk about this on the Tony Robbins Podcast.

Wait, what’s Intensity versus Consistency???

Ok, say I’m looking to get into shape. I want to be physically healthy and fit. Everybody knows that I can’t just go to the gym for 6 hours today and expect to instantly be fit and ripped! No one expects that. That is an example of intensity. Or consider brushing teeth. I can’t just brush my teeth once and hard and then expect my teeth to stay healthy! They’d all fall out after a while! Or let’s say reading books with my kids at bedtime. If I read for two hours before bed tonight with my daughter Lucie it’s not going to make her a better reader. Those are all acts of intensity. Brief intense moments of an event.

We know that we need to brush our teeth twice a day everyday to see results. We know that to go to get in physical shape we need to routinely work out 20 minutes a day and we will see results. We will absolutely get into shape. If we read with our kids every night then they will become better readers and better writers! We know this. These are acts of consistency.

It’s little consistent events that will make the difference not a big one-time or two-time event. But the problem is that we can’t see the benefits of the small acts in real time. I don’t see that my teeth are healthier after the one time brush or that my daughter reads any better…..I have to wait to see those benefits. And it’s hard because those benefits come at different times for different people.

When Simon talks about great leaders he says that great leaders have to build consistency and not intensity. It’s all the little things they do to create a great work culture and not the big hoopla one time event. A great leader can’t just throw an amazing holiday party and expect that to be the solution to a great work environment. They have to show acts everyday that they value their employees. That consistency will create a great work culture!

Intensity versus Consistency for Math Education:

That lesson that flopped was a one-time event! It was because we viewed it as an act of intensity. In order for those lessons to be successful and to bring out deep meaning and learning for our students we need acts of consistency. We need to do this as part of our routine.

That consistency will help create the amazing classroom culture you are dreaming of. That positive, safe, fun learning environment where kids learn with each other and with the teacher! But we have to be willing to put in the work to build consistency.

And the students are not going to be the ones to magically make this happen. We have to do it. We’re talking about middle school or high school here. Students would rather NOT talk to other peers they don’t know. We need to teach them how to help create this environment.

And It’s all the little things we do everyday that will make this happen.

  • It’s the Daily Warm ups where you have kids discussing arguing defending.
  • It’s routinely asking kids to struggle that teach them resilience.
  • It’s your assessment routines.
  • It’s the Random Grouping everyday and using Whiteboards that show them that you value their voice.
  • It’s the problems you use to teach with.
  • It’s how you demonstrate to your students what you value in learning.

It’s the things you do everyday that matter. Those are the things that will build the culture you are looking for. Routine and Consistency are what will drive change in your classroom and student learning.

This takes dedication. I know it’s hard so to help you out I’ve put together a handout that you can download, read and share with other teachers on 5 tips to to build amazing classroom culture. And you guessed it all of them involve being consistent instead of intense.

Click to download 5 Tips to Build Amazing Classroom Culture. 

What are your tips to build amazing classroom culture in your math classes? What are we missing? Just add those in the comments below.

How To Get The Most Out of the Conferences You Attend

Okay wow! You did it. You ponied up for that conference entrance fee, you reserved the hotel, made arrangements for travel! Substitute? Booked! You took a risk! You told yourself I’m a going to that conference! Good for you. Now, you wonder

Is it going to be worth it?

That’s a question I ask all of the time. And my answer now is always yes. And it’s yes as long as I, myself, MAKE it worth it! Great experiences at conferences aren’t just great because we go….they’re great because we made the most out of the experience. My answer to the “worth it question” used to be “I’m not sure it’ll depend on what resources I can get”That’s because my thinking years ago was that the conference and sessions were there to give me that next great activity or project, or worksheet to fill a gap in my unit. And don’t get me wrong conferences are good for that…..but there is so much more to get out of the conferences you attend.

Here are 4 ways to make the most out of the conferences you attend. So that you can always answer yes to the worth it question!

1. Build Your Community.

If you’re like me you’re an introvert. Yes, I shy away from social interaction. I’m that guy at a party standing off to the side with his one or two good friends and avoiding more social interaction. And let’s face it so many of us math teachers are introverts. It’s hard to be social. Going to a conference means we can gather new activities and lesson ideas for our classroom but another huge benefit is Networking. Imagine a group of math educators you could rely on. Imagine you could share your lesson ideas with this group;  get feedback from this group; they even teach like you! Maybe that’s your math department already? Maybe not. A conference is meant and partially designed for you to help create this group. It’s structured for you to easily meet new people who share similar interests. Here are a few tips to help create that group of teachers you need.

Bring a co-worker with you. You’re more likely to meet more people if you are not alone. You’ll have confidence to talk to other small groups. Grab those emails. Learn their twitter handles. You don’t need to do it alone.

Talk at your tables. I know you want to find a table where no one is sitting when you’re at a session but DON’T DO IT. Sit at one with other people. Interact with them. Sitting at the same table already puts you on their team. Now just get to know them. Work together while in that session and get their contact info before they leave if you feel they would make a great team member going further.    

Hit a Social event either before or during the conference. Many conferences will have a games night, trivia night, wine and cheese, or a dinner. Don’t skip on these. This is where so many great connections and friendships can be made. When I attended the annual NCTM conference in San Francisco in 2016 hanging out at Desmos’ games night and the Trivia night were huge aspects of making that conference great. I met so many great teachers that I feel are now apart of my Team!

2. Be Picky

Choose sessions where you recognize the presenter(s). If you recognize a presenters name its likely that you’ve used their activities or ideas from them in the past. By picking those presenters this time it’s highly likely that they will give you more good ideas to go with. When I started choosing sessions based on presenters my whole conference experience changed. It went from “maybe ill get one good idea to try in my classes” to getting a ton of new ideas and left me feeling refreshed and resilient! Even though I had maybe heard them speak before, by choosing them again I always felt rewarded. The conference was definitely “worth it”

I also highly recommend that if you find yourself in a session that you feel is clearly not right for you then leave. Sometimes we feel like it’ll hurt the presenters feelings or you feel that it’s our fault for choosing this session and we have to “stick it out” Don’t! Just get up and leave. And now….sneak into a different session!


3. Check Out The Goods

You probably have filled your schedule up with sessions but don’t forget to budget time to hit the publishers display area. Some conferences have huge value in this area. So many companies get booths just to show off their products to you. You never know what you’ll discover there. While I was at NCTM 2012 in Philly I stumbled across this small booth with this young guy named Eli. He showed me all about this new graphing software I could do on my new iPad or online for Free!! I was in love. Ever since that meeting I’ve been using Desmos in all my math classes. I’ve since gone on and joined a select few to become a Desmos Fellow. So definitely schedule time to check out the publishers area at your conference! It’s worth it.


4. One New Thing

Vow to implement 1 new thing you learn while at the conference. You may get lots of ideas and resources but how many will you actually implement? Thinking about all of those ideas at once can be overwhelming and may result in you not doing any of them. Choose one idea, one resource, one activity, or one routine to try out in your classes when you return. Let’s make it so that you WILL do it. Write down the “one thing” on paper or in your planner. But write it as if it’s past tense, like you’ve already done it. That way it’ll look like you are doing it. For example, if your “one thing” is to try random groupings as a routine then write “I randomize kids everyday when they come into the classroom” By writing it the past tense will help make you do it. Then plan to implement that change routinely. Make a schedule and stick to it. Don’t break the chain. For example, Jerry Seinfeld said that he writes everyday. Not just when he feels inspired or ready to write. He writes everyday. And he focuses on marking each day off on the calendar. A big X right through the day.  As he built up X’s he didn’t think of his goal of writing every day anymore, he just thought, “I can’t break the chain” Breaking the chain meant that he would have to start all over again. Breaking the streak is a more powerful motivator than just “I have to write”.  So set up your goal and then “Don’t Break The Chain”.

Whew! Once I started doing these tips and strategies my conference experience changed. They changed from “I hope it’s worth it” to “This experience is priceless”

Now, there’s a ton of tips listed for you to make your conference worth it above so I created a downloadable PDF for you. It’s a Conference Companion.

You can either print it out or use it digitally on your device. It has places for you to keep important information, like contacts you meet, new ideas, and hashtags. It even has a small scavenger hunt style reminder list along the edges.

Download the Conference Companion.

Go ahead and start making all the conferences you attend worth it!

Do you have tips we can add to this list? Please add a comment below.

How Small Nudges Could Have A Big Impact On Math Education

Do you ever notice that many decisions are secretly being made for us? You probably missed them like I did.

Consider this: Last Friday, and much like every Friday our math department headed out for a quick snack and recap of the week at one of our favourite restaurants. We enjoyed stories from a week’s worth of lessons and working with students while looking forward to new stories for the upcoming weekend. Like every Friday when it’s time to leave I get the bill. What do you notice?

The tip was suggested for me!

I didn’t have to think too hard about what to leave. The restaurant has made it real easy for me to include that good tip. How many more tips do you think this restaurant earns compared to other restaurants that don’t have this feature? A bunch more is the answer.  

Here’s another example of a subtle suggestion that has a big influence on our decisions. Have a look at this image taken at the University of Pennsylvania.  

We have to do a double take and think about what side to place our trash in. We are pushed to consider our trash placement instead of just tossing it in a bin. A subtle suggestion that influences our decision. Recycle or Landfill?

Both of these scenarios didn’t just happen by accident or without careful thought, both are using a branch of behavioural economics to influence decision making called Nudge Theory.

From wikipedia: “A Nudge is a concept in behavioural science, political theory and economics which proposes positive reinforcement and indirect suggestions as ways to influence the behavior and decision making of groups or individuals.”

If we go back to the tipping example: the suggested tip amount on the bill is a small indirect suggestion for me to give a good tip. The restaurant has made it real easy to leave that tip (they did the math for me). Also notice that they didn’t provide tip amounts less than 15%. They nudged me to make a tip of 15% or greater! They nudged me to choose the behaviour that was favourable to them.

In the garbage bin example the creators drew your attention to the negative impacts of putting items in the garbage side. If you toss on the garbage side that trash goes directly to the landfill. They nudged you, very subtly, to think before you toss trash.

The basic idea behind a successful nudge is to make it very easy to do the favourable behaviour and hard to do the unfavourable one. It was easy for me to make a tip of 20% and harder if I wanted to tip less. It was easy for us to make a decision to recycle because choosing the alternative was something we generally want to avoid: filling landfills.

Nobel Prize winning economist Richard Thaler along with Cass Sunstein brought Nudges into the forefront in 2008 with their book: Nudge: Improving Decisions about Health, Wealth, and Happiness and I learned about it from the Choiceology Podcast episode: The Big Impact of Small Changes by Dan Heath. In that episode Heath outlines a great example of a nudge: the power of the opt-in versus the opt-out.

In Austria, more than 90% of their citizens are organ donors. In the neighbouring country Germany? Less than 15%. Are Austrians that much more conscience about organ donation? Nope. In Austria citizens are auto-enrolled in the program and have to opt-out if they would rather not be a donor. In Germany you are not auto-enrolled and have to opt-in to be a donor. That’s a nudge! A small subtle change can have big outcomes in decisions. Auto-enrolling capitalizes on our laziness factor. It’s easier if we do nothing compared to doing something.

“It’s a question of putting the best outcome along the path of least resistance and letting your automatic system do its thing.” Mark W Riepe writes regarding nudges.

What implications does Nudge Theory have in the math classroom?

 Nudging in Math Education

In my opinion one of the most poorly designed calculator features on my smartphone is the percent button.

Students often misuse the % button the calculator and are not sure what is going one behind that it. It’s like a black box of percent calculation. Many misconceptions arise in my grade 9 applied class when students use this button without knowing the math behind it.

Compare that to what Desmos is doing.

Desmos has auto-added the word “of” after the percent symbol. There is no way for a student to delete the word of. It’s stuck there. It forces us to think about how we use that button. Both teachers and students are now auto-enrolled in finding the percent of a number before doing anything else. It’s an amazingly small change that has a huge impact on learning. They have made it easy to do the right thing. And impossible to do the wrong thing. That is a nudge!

Have you seen these charging stations for phones in classrooms?

Teachers are making it super easy for students to choose to put their phone away if it’s a distraction to their learning. A student will gladly get the juice and put that phone away for an hour. Nudge!

I want students to regularly self assess their learning in my classroom. I want them to be more accountable to get what they need. I make it very easy for them to see their own progress on the learning goals in our course. By auto-enrolling them in their Freshgrade portfolios student can access all the learning goals anytime and work towards showing improvement on them! It’s my way of giving them a nudge to make good decisions regarding their assessment.

What kind of nudges are you trying?

What are you doing to make the right behaviours easy and the wrong ones hard? What are you doing to affect choices your students make in your classroom?