A Look at the Mathematical Reasoning of Third Graders....hmmmm?

 Yesterday, I gave my kiddos a problem to ponder as they came into the room.  It seemed pretty straight forward, but I knew it would take some thinking, deciding, and it was multiple steps so there was a potential for confusion.  The kids settled into their thinking pretty decently; considering the time of the year.

I can't remember where I found the problem, but it went something like this:

Bredan and Katie are playing Monopoly.  At the end of the game Bredan has $120 more than Katie.  Together they have $840.  How much  did each child have?

WELL, I could tell it was giving the kids a real challenge and I encouraged them to use strategies we've used throughout the year.  After about 15 minutes I asked them to hand in their papers and let them know we'd take a deeper look.

When I looked through the solutions and the array of answers, I decided we needed to spend more time thinking on this concept.  I put together a chart of the answers.  On this page I also put down the basic facts from the problem; a total of $840 and Bredan had $120 more than Katie.  I have 19 kids in my class, so only a few had the same answer and some didn't even have one yet.

These are the answers the class came up with on the first day.

 I gave them the challenge to look over all the answers, re-examine their work and make some decisions.  Do you want to stick with your answer and defend it or change to another answer and decide why?  I let the kids know they could work as long as necessary, but to make sure to circle their answer in a bright color.

During lunch, I looked over their work and recorded their decisions on the chart from the morning.  When we had a bit of time in the afternoon we looked over the work and tried to prove each of the answers.  It's fascinating to hear what they are thinking.  Most of the answers now totaled up to $840 but the method to get to a $120 difference was baffling to many of the kiddos.  Some of the kids split the money up equally and took $120 from Katie and gave it to Bredan.  They didn't understand that now Bredan has $240 more than Katie.


Earlier this year they got confused by a problem with equal groups and changing to a group that had 2 more than the other.

There are 4 kids in each group.  If you need to change the groups so there are 2 more in one of the groups, how would you do it?

 Many of the kids thought you would move 2 kids into the other group.  Of course, you see that now one group has 2 kids and the other has 6.

This is the work we did with the problem.  First we added up each one that was starred in black.  If it added up to $840 we kept it in the list, if not, we crossed it out.  After that we painfully came up with how we would find out if Bredan had $120 more.  We crossed out any that didn't have a difference of $120.  Amazingly enough, we ended up with one answer that worked for both!!

The lesson in all this it NOT that I'm a terrible teacher, but that children have misconceptions.  They won't be able to reason for themselves until we peel away the layers and guide them to some DEEP thinking.  DIGGING INTO LEARNING...with LOTS of patience!!!