How do we deepen place value understanding when using the standard algorithm for addition?

This week I joined some fourth graders as they tackled missing digit addition problems.  Prior to this lesson, they had spent a great deal of time on both their place value understanding and strengthening their use of the standard algorithm for addition.  The new goal was to apply this place value understanding as they added multi-digit numbers using the standard algorithm.  I was in awe over how they persevered.

To start the class, the teacher asked the students how they felt about the homework that had done the night before, which was an introduction to the missing digit problems that are highlighted below.  Most comments were negative:

• "I struggled with it at first."  
• "It was challenging without the teacher to help me."  
• "It felt tricky...when I had to carry."  
• One student said that he tried and put a mark where he was unsure so he could ask today.

What raw honesty from these mathematicians!  We are three weeks into school and they felt secure enough to share their challenges.  Bravo to the classroom teacher for establishing such a nurturing and warm classroom community.

The teacher then selected several problems for the students to grapple with, starting with a simpler 2-digit by 2-digit addition problem.  The students worked to solve it on white boards.  Yes, white boards are an engaging tool and it's so easy to erase when you change your thinking.
                                                     

    The teacher certainly facilitated the discussion, but all the thinking came from the students.  They verbalized how they needed one more in the ones place to make eight and some had to pause to think about how they could make thirteen tens (or three tens and one hundred).

The next problem upped the ante.

They shared their ideas around place value and the need to "carry" one group of ten , if they had 17 ones.  As students shared their thinking, other students took it all in.  At this point the student who had originally marked where he got stuck on his homework, raised his hand to share.  YES!

Then they moved onto three digit numbers.  In this situation, three different answers surfaced among the students.  The white boards offered instant feedback on who was challenged with carrying a group of ten.

We can all learn from misconceptions.

Students shared how they knew 6 + 4 = 10, but were still not sure how to show that because they were not considering that they may also need to continue to carry to the hundreds place.  The idea of adding your own box to carry resurfaced.  Another idea that came up was checking your work with subtraction.  The students were justifying their thinking and learning from each other.

When the students went off to work independently, the task involved adding four and five digit numbers with missing digits.  No student wavered.  They were deeply thinking about place value and some continued to try out their thinking on the white board before putting the pencil to paper.

As I observed students working, I notice two students had made different errors on the same problem.  This was a perfect opportunity for them to continue to learn together.  The two students talked about the place value of each digit and both "discovered" their errors as they explained their thinking. They were really taking ownership of their learning.

As the teacher gathered students back together to share their new learning, the energy and tone of the class had completely changed.  Many felt empowered by working through this struggle.  In turn, they had deepened their understanding of place value.

In the end the teacher reminded them, that when they learn something new, it can be hard, but to stop and think, "I may not understand it yet."  I loved the word, "YET" - this is the growth mindset of this classroom.

What are some ways you offer your students a productive struggle as they deepen their understanding of place value.  Share your ideas.  We learn from each other.