Exploding Dots and other Math Circle success

This summer I attended a Math Circles workshop.  I wasn't really sure what to expect, but since I can always use new math ideas I signed up.  You know what happened when we got there?  They made us do math!!  Yup. They did.  Lots and lots of math.

They made us do math that wasn't in our areas.  I had to go back to my high school years to remember some of this stuff!  I probably could have used my university years, but I blocked out all of that math.


I did math I'd never heard of!  We played a game, and the instructions included the word "hyperplane."  What on earth?!  Then, we worked on this thing called a hypercube.  I'm still not entirely sure what those things are, but you know what?  It's okay.  It's okay not to know all the things.  You are still a math person if you don't know all the things.

So, why am I putting this on my classroom blog?  I've spent years wondering why exactly my students struggle so much with place value.  It's so easy!  You go from one spot to the next, and the values change in a predictable manner.  The digits change in a predictable manner.  What was so hard?  You know what I found out at Math Circle?  Place value is hard.  We've just been doing it so long that it's easy and natural for us.  My favorite professional development is the one that can make the things that are easy for me hard - just like it is for my students.

"Well," you ask, "what was it that made you think place value was hard?  I'm so glad you asked!  Exploding Dots.  Yup.  You read it correctly.  We exploded dots.  How many of you know this fact:  our number system is based on tens?  Or have to teach this one:  we use a base 10 system?  Most of us would say we do.  It's why we focus so much on base 10 place value.  It's why we in elementary schools search frantically trying to find inexpensive Base 10 Blocks so we can have enough for all of our students to use.  But you know what?  Those are not the entire truth.  We use base 10 for computations.  Computers use base 2 (otherwise known as binary) to do all of their work.  Measurement changes every 3 feet to 1 yard, every 8 ounces to 1 cup.  Time changes every 24 hours to one day, every 60 minutes to one hour.  Then there is money!  Four quarters to 1 dollar, but 10 dimes to that same 1 dollar.  And yet we wonder why place value is so hard for kids! They have to memorize all of these exchanges, and then to do anything computational with them, they combine those exchanges with the base 10 values.  Basic math can be complicated!

What happens if we don't start with base 10?  What if we just start with this idea that you can create a system with predictable patterns by exchanging one group of things for another?  That's what exploding dots does.  In our math circle, we did all kinds of complicated things with it, but I teach 2nd graders.  I just needed the basics.  I started with base two.  I taught them how to exchange two dots for one and to move them over to the next place on the chart.  We created "secret codes" for the numbers 1 - 10.  It was tough in the beginning.  They didn't quite get it, and it took a lot of discussion to get it right.  But in the end, we had our code.

Then I moved on to base 3.  They got this one faster and with much less help from me.  It was fascinating to watch them as they started teaching each other.  Then they acted it out, and that really helped some of them grasp what was going on.  Once our base 3 code was complete, I gave them the challenge of base 4. 

This one they got in less than one class period.  I didn't help them work out a single answer.  Not one.  They got the entire code correct by putting their answers up on the board and fixing their own errors.

All right, but what does that mean for their learning?  How do I know this helped?  This was my own adaptation from our math circle exercise, but I wanted to see what happened.  I wrote all of our codes down on the board, and I added a column. 

I gave them a challenge.  I pointed to the information on the board, and I told them they had 15 minutes to discuss with their group what they thought would happen if I said they now had a 10 to 1 exchange.  What would the code look like?  I didn't give them paper to work on or any manipulatives.  I wanted to see what they would say.

It took them less than 5 minutes.  Less than 5 minutes!  I had 24 kids talking in their groups, completely on task, and explaining to their partners why their answer was correct.  Hands were popping up all over the room to share their answer.  So I gave them another challenge.  I bet them.  I said if one person in the room was confident enough to take the bet, and they were correct, I'd give them all free drawing time or Go Noodle for 15 minutes.  If they were incorrect, they got the consequence of my choice.  Now, keep in mind, I'd heard the discussions.  I knew there were correct answers.  Fewer hands popped up, but I still got a good chunk.  I called on the first 2 I saw, and one young lady that I knew had the right answer but hadn't had the courage to raise her hand.  I let them get together to see if they had the same answers (I knew they did) and to decide if they were going to take my bet.

Here is what I got.

Did they learn what I taught?  You bet.  Am I confident that they are going to understand place value?  Yup.  I am.  Do I have one young lady that will have more confidence next time?  Sure do.  The value part of place value will be tricky as always, but look at the brains I have to work with!