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.

Native Village Simulation

Lately, it's seems that science and social studies gets the short end of the straw.  The CORE is all about writing, reading and math.  Kids are working harder than ever, pushing themselves to go deep and yet they are STILL kids.  They like to have fun, experience parts of history and the world they have no clue about...am I right?

In 3rd grade we learn about Native American cultures; linking the learning with the region.  It's one of my favorite units since I can incorporate the habitats and natural resources throughout the unit.  This year our time for social studies and science has been quite condensed to say the least.  Usually we use books and the Internet to research and then take time creating art, food and playing games from each region.  This year I couldn't cross over into literacy for research due to our new curriculum soooooooo....I had to get creative.

Another 3rd grade teacher and I were each in the same situation, but her worries were in the colonial time period.  Well, two heads are indeed better than one, and we came up with a fabulous solution.  We would take a Friday to celebrate our learning.  Her room would be a colonial village and mine would be a native American village from the east coast during the same time period.  Planning what we would include was inspiring and reminded me why I love to teach!!

The day before was a ton of setting up, but well worth the look on the kids faces as they came into the room.  One area of the room was set up for learning about food and doing some sampling.  The couch area was the fire circle where the village would hear stories and work on their leather projects.  Another area was set aside to learn about birch bark baskets.  Finally the challenging part was creating a wigwam.  I was hoping to get some enthusiastic, creative parents - but that didn't quite happen.  I envisioned the wigwam for kids to read and work on weaving.  I had the bright idea that a screen porch structure might work......the one I had didn't have directions and I struggled for about an hour to put it together (I'm not very "build it" smart).  Once it was together I realized it was HUGE and not quite my vision.  I took half the pieces off and built a smaller version.  The legs needed bracing and I threw the screen tarp over the top and then put on the "bark".  I must say it turned out pretty close to my idea and was glad I put the time into it.

The day of the simulation we divided up our kids with half and half going to each room so there was a bit of a mix.  I had great volunteers.  One parent and a grandparent designed the birch bark project - made with paper.  They also ran this station.  Another volunteer who knows my classroom really well floated and helped kids our if they finished an area early.  I have an amazing community volunteer who loves to teach children about food.  She focused on corn, beans and squash.  She showed the kids the uncooked version of each.  They used a mortar and pestle to grind corn and took dried corn off the stalk.  All the kids sampled corn muffins, bean soup and a bit of squash.  I had the fun of showing kids moccasins made from leather and teaching about using hides.  Each of the villages made a little pouch from deer hides that my father-in-law found at a rummage sale.  It was a favorite project of many.

The kids learned so much, asked wonderful questions and were VERY appreciative.  Simulations bring to life the learning,  foster connections and build memories for a lifetime!!

Math Discussions Build Knowledge and Community

The CORE is filled with pieces related to discourse, discussions, critiquing and analyzing our work.  Let me tell you this can be a scary thing for a kid or an adult.  Can you imagine putting your work up in front of a group to critique?  You MUST have a community that is built around sharing, discussing and acceptance.  The idea is for children to be able to share and learn from their mistakes.  It's quite amazing to see the lightbulbs go on as the group analyzes the work.

When we start out our math workshop I usually have the kids start working on a problem while they munch on their snack.  On Friday I thought the problem was pretty straight forward...famous last words.

The problem stated:

Sarah is getting invitations ready for her birthday party.  She has 10 invitations.  She puts the invitations in two piles.  One pile has 2 more invitations than the other pile.  How many invitations are in each pile?

I share the first few on the screen so we can discuss problem solving strategies.  The first four answers were; 3 and 7, 4 and 6, 2 and 8 and 3 and 7.  Not as easy as I imagined and the thought process and reasoning was interesting.  After each child shares their reasoning the class is asked to comment or question?  Our community of learners has grown so much this year - they were respectful and asked wonderful questions.  Some of the kids even demonstrated their thinking with manipulatives.

After this I had the groups divide up into their answer they would like to defend.  Most of the class was in the group of 4 and 6, while a few kids were sticking with their ideas of each pile at 5 and 5 - you take 2 from the pile, so now it's 2 bigger 7 and 3.  Fascinating!!

Each group had the opportunity to share their ideas and to prove their thinking.  They were also told that at any time they could change their opinion and move to the other group.  I let the 3 and 7 kids go first.  They didn't convince anyone to change to their side.

Next the 4 and 6 kids went.  They planned to demonstrate their thinking with kids lined up (since it was such a larger group).  Demonstrating that with 6 in one line and 4 in the other - there were 2 more kids in the one line, just like the problem.  I could see the kids in the other group were getting the idea, but they weren't ready to change their thinking yet!!!

I brought out some small whiteboards, sort of envelope like.  I started with a number that wasn't 10.  So if I have 6 whiteboards and I put 2 in one pile and 4 in another pile, how many more are in this pile?  The kids can see it's 2 more.  So I  take 8 whiteboards.  If I have 5 in one pile and 3 in another pile, how many more are in this pile?  Finally I take out 10 whiteboards.  I put together different combinations, always asking how many more in this pile?  Whew, we finally got to the point.

The most awesome part of it was how the kids in the community responded and reacted to one another.  They were respectful and patient.  The learning of the entire community was moved along by digging into the misunderstandings of a few.  Life lessons were learned about communicating with others, questioning and defending your thinking.