To model or not to model. That is the question.

This is the problem I started with on the second Monday of 3rd grade:

Mrs. Cleveland found a bunch of pencils in her garage. She organized them into bags and boxes. Each bag could hold 10 pencils and each box could hold 10 bags. If she had 3 full boxes and 7 full bags how many pencils did she have? 

Multiplication you say?  Place value I say.  I needed to assess their place value knowledge.  What did I find?  Nearly half my students showed evidence that they knew 10 tens is 100.  Many of the other students had good counting strategies to figure it out.  I did have a few students who were unsuccessful but only three who didn't have a place to start.  I counted it as a success.  The next day's discussion revealed that even some of the students who didn't show evidence of place value understanding when solving the problem actually did have some understanding, they just got lost in the steps of solving this particular problem. 



The next problem I posed was a measurement division problem using groups of ten.  There were 270 pieces of candy being put into goodie bags.  Each bag was to hold 10 pieces of candy.  The students were asked to find out how many goodie bags could be made.  With this problem, the same group of kids that were unsuccessful the day before were unsuccessful again plus a few.  The difference was that this time, most of them didn't even have a valid strategy. Perhaps they weren't ready for this type of problem, or perhaps I didn't support them enough in making sense of the problem.  The day before we had done a little skit and acted out putting pencils in bags and bags in boxes (though we didn't do any solving together.)  On the other end of the spectrum I had 6 students who wrote some sort of multiplication number sentence such as 27 x 10 = 270 or a string of multiplication sentences to arrive at the total. 

After Classroom Discussion

With the variety of strategies my students used, I felt like I had the perfect papers to introduce multiplication in our next discussion, but I didn't.  Mostly because I was worried about those students who weren't making sense of groups yet and weren't using grouping strategies on these problems.  Instead I shared a valid strategy and represented all 27 bags but displayed an answer of 26 bags.  We focused on the standard for mathematical practice 6: attend to precision as well as grouping.  We circled groups within the strategy to double check the answer and realized it was written incorrectly.  
As a side note, I was careful not to have the students go back and count one by one to check the work, because that encourages the most inefficient strategy... counting by ones.

After our discussion about grouping, I decided to return to a multiplication problem but this time with predetermined groups of 100.  I was hoping the students wouldn't get caught up on the multiple steps aspect of the problem solving.  The problem I posed was this:

All of MMJ students are going on a field trip to the Zoo.  The buses will hold 100 students and the vans will hold 10 students.  If there are 6 full buses and 4 full vans then how many students will go on the field trip?

Direct Place Value

Success!  19 of my 23 students were completely successful.  Many of the kids were still using a counting strategy, but most of them were writing repeated addition sentences.  Almost all of the students grouped their hundreds together and then the tens when writing these sentences.  This is likely due to the structure of the problem, but it gave me a great platform to introduce the multiplication symbol.  Again I had several students who relied on direct place value understanding to solve the problem.  I even had 3 students who had not yet used place value understanding move to using place value strategies in this problem.  I was excited about the progress.  

On Thursday, I decided that I would pose one more problem before jumping into introducing the multiplication symbol.  Because of the success on the most recent problem, I decided to take problem solving one step further and introduce multiple number sets.  This, I feel, is essential to differentiation.  I have a rather large group of students who have direct place value understanding and I needed to push their thinking.  So I chose to make my third number set 18 groups of 5 and wondered if any of them would make the connection back to the groups of 10 that we had been working with. 


With the implementation of multiple number sets, I felt like I needed to model for my students.  They needed to be shown how to plug in the numbers and keep their papers organized.  So I used a basic problem with blanks and number sets and we worked together as a class to go through the process.  First we entered the numbers from the first set and solved the problem.  We discussed how they should show what they did with either a picture or a number sentence or both.  I also pointed out the need for labels and showed them how to draw a line to separate the work for each number set.  Then we moved on the next set of numbers and went through the process again.  After solving the problem 3 times, we did a brain break (www.gonoodle.com).  They needed it!

I then posed a multiplication problem with 3 number sets: 8 groups of 10, 18 groups of 10 and 18 groups of 5.  After a quick story telling session and visualization of the problem I sent my students off to their quiet spots to solve.  You will not guess what happened next.  They ADDED!  I looked at the first completed problem and said, "Would you please go draw me a picture that matches the story."  Then another student showed me the same strategy.  Student after student brought their paper to me with three addition sentences and exclaimed, "I'm done."  Nearly half my class.  These were students that had been using grouping strategies and place value understanding just the day before.  What happened?

I modeled.  I showed them how to solve the problem.  Well not really, but really.  Unintentionally, while modeling for them how to solve with multiple number sets, I showed them how to solve a part part whole problem with addition and they took what I showed them and applied it to the new problem.  I was in shock.  All week these students had been problem solving and progressing.  Several went from not having a strategy to having a viable strategy to being successful.  The minute I provided them with direct teaching, no matter what my intent, they latched on and all sense making went out the window. 

Granted, not every student regressed.  As a matter of fact, even many of the students who added revised their strategy when asked to draw a picture to match the problem and support their number sentence.  Still though, initially they adopted my strategy and thought nothing of it.  So hear me out, I believe that students have an inate problem solving ability.  When given the opportunity to think through a problem on their own and make sense, they can do it.  Some will make sense faster than others, but still they can.  They do not need us to show them how to solve the problem, they need us to facilitate their thinking and question them so that they make connections.  At least for a certain population of our students, modeling and mini-lessons impair their own problem solving abilities.  If you want multiple strategies, don't show them yours.