# 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/

# Building Resilient & Determined Math Students

Are you frustrated with how easily some of your students just give up while doing a math problem? You know that if they just stick with it that they will learn but they just want to be hand-held through math class every day. In the book How Children Succeed: Grit, Curiosity and the hidden power of character  Paul Tough argues that students succeed not because of intelligence but because of how much stick-to-it-ness, grit, and Determination they have.

It’s not that I’m so smart, it’s just that I stay with problems longer. – Albert Einstein.

Tough says that you can build perseverance in children by playing chess. From the book, “Teaching chess is really about teaching the habits that go along with thinking,” Spiegel explained to me one morning when I visited her classroom. “Like how to understand your mistakes and how to be more aware of your thought processes.” Playing chess over and over builds up a chess player’s level of determination. They have to take risks and learn from those risks in order to succeed. If we want our math students to build up resilience and determination then we also have to push them take risks and learn from the outcome of those risks.

In math class we can build up resilience, grit and stick-to-it-ness if we put students in experiences where they have to persevere through a tough situation. But think of their whole math class experience up to this point. It’s likely that a student would  never have had the opportunity to try to solve a problem before we math teachers show them the examples and how to solve it the math teacher way. Our students need experience persevering through tough situations like the chess player.

Imagine the first time you play chess and your opponent takes your bishop early in the game. You might think the game is pretty much over. Why go on? Or think of the young basketball player who has the right footing for a layup. They definitely weren’t a pro at that the first few times. But over time in each situation players overcome that resistance and persevere. They learn to be successful.

But in math class we assume math students should be good problem solvers and have grit in our math classes immediately. We say “our students give up too quickly” but when did we ever give them time to build those perseverance skills up? When did we teach them how to persevere? We are the ones that have to give them experiences to build that skill up.

## 3 Tips to Prevent the “Give Up Moments” and Create resilient Problem Solvers

1. Routinely have students solve unfamiliar problems through a supportive productive struggle process.

Use the Hero’s Journey to structure your math class and create productive struggle moments daily for your students. As an example, if I didn’t push my students to solve these problems routinely on their own to start our lesson then they would not only miss gaining the experience to persevere

but I the teacher would also miss gaining valuable information about what my students know or don’t know. Problem solving must be a regular part of learning not just a once a unit or end of unit thing.

2. Create an environment where risk taking is low stakes.

In order for students to take risks and learn how to persevere the stakes for failure have to be low. It has to be painless to make mistakes. How are we doing this in our math classes? One easy-to-implement technique to make risk-taking low stakes is to bring dry-erase boards into your classroom. The no-permanence of the boards makes risk taking easy and it’s one of my favourite things. Students can attempt strategies quickly and wipe away quickly if needed. You can read more about the research behind non-permanent surfaces from Peter Liljedahl.

3. Show students that you value perseverance:

Create an assessment routine that promotes growth instead grades. Students quickly learn what you value. If we’re saying to them daily that we value the process of their learning over the final answer then how to we prove it to them? Your actions speak loudly. Give your students room to show that they have persevered while solving problems. Learn how you can implement an assessment routine that promotes growth and resilience by watching Conall’s Assessment story:

Disclaimer: This transformation won’t happen over night. You yourself have to be resilient and determined. It’s possible that you might not see that change even this semester. But by allowing students to productively struggle through problems, giving them a low stakes risk taking environment and proving to them you value persistence WILL build their resilience and determination in the long term. We also must have a stick-to-it-ness to build great thinkers!

# 4 Ways To Use Two Truths & One Lie in ANY Math Class.

Two truths and one lie in math class has taken math class discussions and math arguments to a whole new level! They are the main event when it comes to math fights!

Watch the video post:

Not into watching the video? No problem, read the transcript below.