What makes students remember the math they are learning? Is it because you’re using a real world problem that they can relate to? Is it because maybe you used a 3-Act task? Is it because they practiced the content over and over? Is it because you used spaced practice versus massed practice? My good friend Kyle Pearce and I believe it is much more than that.

While at Oame 2018 Kyle and I took a chance and hit record on Facebook Live during our 75 minute workshop title Going Deeper with Math Moments That Matter. If you missed it or want to learn more you can watch the whole thing right here!

Session Description:

What makes a memorable math moment? Is it a real world task? Is it relevant to your students? Is it media-rich or delivered in 3 acts? While many professional development sessions focus on a specific component of an effective math lesson, Jon Orr and Kyle Pearce will model what they believe to be the three key components of an effective mathematics lesson: sparking student curiosity, fuelling their sense making and igniting your next steps. Join them as they lead a task to break each component down and then build it all back up to create a memorable math moment.

What were your moments that you remember from math class?

What do you want your students to remember 5 years from now? Leave comments below. Or jump over to my Facebook Group and you can comment there.

Grab the Making Math Moments Matter Curious Task Template and our file with support resources over at makemathmoments.com

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 mathematicalgame 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!

I said nothing else. I was letting the curiosity build. After it looped and looped students started to work. Without saying a word about it students were trying to find how long it will take to defrost an item that weighed 3.5 pounds.

Igniting MY Moves:

Since I routinely let students struggle to solve problems instead of showing them immediately a “how to”, I have to be ready to give feedback on what they try on the fly. I want to help push them in not only a direction that solves the problem but prepares them to see solutions that are not their own and solutions that attempt to address our learning goal for that day. That takes careful planning which is not an easy thing.

“Plan with Precision so you can proceed with great flexibility” – Tom Schimmer.

When I first started teaching so much of my planning was solely focused on answering questions like, What topic? What examples?, and How long do I spend on it? Now my planning time is mostly spent trying to answer: How will the students solve this problem? How can I use what they will do to shape the lesson? What do their attempted solutions tell me about what they have learned so far? So my planning process has gone from examples like this where I was so concerned with WHAT….

to spending most of my time thinking about HOW. HOW will the students respond to the task? What does that look like? That takes a ton of anticipation. Anticipating their solutions and strategies puts me in a better position to understand their thinking and help shape that thinking. For each possible attempt I need to be ready to provide feedback to help them achieve our goals.

For the Defrost Black Box problem from above the learning goal I am hoping to pull out is “Relations can be represented in various ways” and “Problems can be solved in a variety of ways” I anticipated that some of my students would attempt to solve it with a unit rate.

Possibly some of my students may solve it with a table of values and linear relation.

Since my goal is for students to see “Relations can be represented in various ways” and “Problems can be solved in a variety of ways”

I need to be ready to fuel their sense making by linking the different student strategies together.

Here are some of what the students tried.

I did not anticipate students using seconds.

I also did not anticipate students using additive thinking with the unit rate.

We learn so much from our students by allowing them to show their thinking. Imagine all the missed conversations with my students from 2005 – 2013. Imagine how many of my students felt like they were failures because their brains didn’t tell them to solve those problems the same way the I did. When in reality they had so many good insights that just needed to be tailored.

Selected students presented their strategies to the class. Now it was time to show how their strategies connect together.

We showed how the unit rates that many of them found and used showed up the table solution.

We moved from there to show how this would be represented on our number lines.

Yes the planning that comes from Igniting My Moves and Fuel Sense Making takes time and it is not easy. But I can tell you that it is worth it.

When it comes to angles involving parallel lines, triangles, and other polygons I’ve always assumed my grade 9 applied students “get this”. I’ve felt that angles were an easy topic. I guess I thought this because most students seem pretty happy when solving angle problems and for the most part being doing pretty well on assessments. However, this year I noticed two inadequacies that I am trying to address.

Most of my students didn’t actually know what an angle measurement of 65 degrees really means.

They have a hard time determining what information is needed when solving multi-step angle problems. Lack of a good strategy.

Addressing #1

When having students determine angles in triangles almost all of them knew that all three angles should add to 180 degrees. The trouble came when I saw some answers like this (from more than one student).

What bothered me was the location of the 40. I wondered why outside the triangle? I pressed this student for more info. I asked him to draw me any right triangle and label the three angles.

Hmmm…I asked him to point to one of the angles. He pointed to where he labeled the 85. What I found is that this student was mixing up length measurements with rotational measurements and he was not alone.

I found a great activity to hit this head on. Laser Challenge from Desmos worked wonders to get my students to understand and experience rotational measurements. Students have to enter values to rotate the laser and mirror to hit targets.

My students “felt” what 60 degrees is. Experiencing that rotation made all the difference to clear up what we were actually measuring. When second semester rolled around and my new crop of kids came in we started with this activity right away.

Addressing #2

Most of our students struggle with solving complex problems where they have to think of a strategy. Before I gave them something like this,

I wanted to them to experience what information would be useful to know first. I decided to turn the problem around and inside out.

I gave them this.

I wanted them to think backwards….just like we need to do sometimes when solving longer problems. On the “easy” side most filled in 3 angles in the quadrilateral. What was great was that prepared them to think what we could leave out for the harder one. This simpler diagram challenged my class to think, plan, and strategize!

It was great to do this before we introduced this puzzle Jim Roesch, Kristyn Wilson, and myself created:

And to really challenge yourself or your students here is a blank one. Can you fill it in so it’s “hard” to determine that indicated angle? What is the least amount of info you can give to bring out the most amount of thinking? Share them out!