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!!!

Questioning in Science: A Study in Trees

“It is not that I'm so smart. But I stay with the questions much longer.” 
― Albert Einstein

It seems as though in education these days we are always in a hurry, trying to fit in more and more.  It may be wise to look to the past for inspiration.  This quote by Albert Einstein is certainly worth reflection.  In the CORE and the new NGSS - Next Generation Science Standards the idea is to get children to slow down, think and question.  The other day I asked my kids to come up with some questions about some tree branches that I'd gathered.  They came up with some research questions, but struggled to come up with things they could discover without books.

My challenge as a teacher is to help children to question and wonder.  We want our kids to actively explore, think and reason.  If we constantly plan out every detail for the kids and give them the "correct" answer after an experiment, how will they every learn to solve problems, invent and innovate?

So I tried again by putting out tree branches, cones, flowers and buds in the room.  I set up some microscopes and magnifiers.  Still the kids weren't taking the bait, yes a few looked and said wow, but no good questions or discoveries.

I decided we needed a trek outdoors to gather their own research material.  Of course it didn't hurt that it was a lovely day!  Now they were enthusiastic, maybe not for the reasons I wanted, but at least there was some excitement.  They brought their leaf back to the room, opened their science notebooks and started to draw and write their observations.

After this we gathered in the living room to chart what we noticed.  The kiddos had some intriguing discoveries.  I charted their ideas in a somewhat organized way.  If someone said their leaf was fuzzy, I asked if that was true for all of them.  Our list grew into things that were different and some that were all the same.  One property in common was "veins" or "tunnels" in all the leaves.  I threw out a question, "What is the job of a leave?  Why do trees have leaves?"  We webbed out some of their ideas on the chart.  How do you think we could prove our ideas?

Now the group was humming.  They came up with lots of ideas and we decided to try putting food coloring into some water.  The kids volunteered their leaves as test subjects.  Another idea was to look at the make-up of the leaf under the microscope to see the veins better.  They wanted to know if the vein was hollow or solid.  I wonder where their thinking will go if they discover one or the other?

One particularly observant child, who spends lots of time in the woods, shares a bit of knowledge.  He broke open a leaf once and "water" came out of the vein.  Hmmmm this really has the group buzzing now.  Science in the next few years is going to change.  Thinking, questioning and designing will be a part of our everyday learning.  Kids will need a nudge and a bit of a bump to get their thinking going, but I can't wait to see the results.

Math Notebook Routines

Over the years I have loved the writer's notebook and in the past few years have gotten into using a reader's notebook, but I haven't used a math notebook.  It's hard to explain...but I just didn't.  Recently I've been reading a lot about mathematical thinking and learning.  Last summer I took Math Frameworks and we used Van de Walle Professional Math Series 3-5.

http://www.ablongman.com/vandewalleseries/

Our curriculum director is always on the lookout for good books and trends in education.  She offered a study group for 4th-8th grade teachers.  I weaseled my way in because I usually loop with my kids to 4th grade.  Minds on Mathematics: Using Math Workshop to Develop Deep Understanding in Grades 4-8 was the book to be "studied".  The group of teachers I worked with and the book helped to stimulate my thinking and questioning.  I came to understand that the math notebook helps children to process their thinking and to deepen their understanding.  In the CORE deeper understanding and explaining our thinking is a huge piece.
Minds on Mathematics
There are lots of ways to use a math notebook in the classroom, but routines with the notebook are important.  Each day when we come back from specials, it's snack time.  Time in the classroom is a precious commodity, so we have to use every minute.  When the kiddos bop into the room, there is a problem sitting out at their desk.  They get out their notebook, glue stick and pencil and get to work.  Our routines are in place for a reason, their brain can get thinking on the problem without any directions at all.  The bonus is they get to munch on their snack as they work.  Just to make sure they get going right away we have a scheduled time to start on the mini-lesson.

I find problems all over the place and frequently create my own.  I just put a problem set on Teacher's Pay Teacher's.  It has end of the year and beginning of summer type problems.  I like to keep the thinking going right up until the end and I must admit the kids like problems about SUMMER.