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.

                                               

Fostering the "Can do!" Attitude

The other day I brought my daughter to college.  This leaves my husband and I labeled as "empty nesters".  Several years back when her brother went off to college, I ended up on blood pressure medicine.  So I was expecting this move to be rough to say the least.

I am thrilled to report that this time around it was an amazing experience!  I embraced the day, savoring the countless opportunities I saw unfold before us.  To my complete surprise, for the last couple days, rather than looking in her room with sadness, I have been filled with such a feeling of joy for her.

Those of you that know me are thinking, "Is this Lori talking?"  I have always been the person that does not welcome change.  This new found thinking really caused me to pause and reflect about how it came to be.

I can honestly say that the change in my mindset has evolved over time - a real process.  What influenced me?  I am surrounded by some dear friends and colleagues that have showed me how to welcome these new experiences.  I guess you could say that they've really rubbed off on me.

This brought me to thinking about kids who say or think, "I'm not good at math."  We all have them.  Heck, we have adults around us singing the same tune.  So how do we influence our students' mindset?

In my blog post, Tackling the Summer Slide, I discuss encouraging young mathematicians by using the language, "I'm still learning to..."  As I have worked with students who have tried out this phrase, I could actually hear a lighter tone in their voices as they say it.  They feel empowered to have permission to express that they are working on their learning, rather than having a feeling a defeat.  But what else can we do as educators to foster this growth mindset?

Making sense of problems and persevere in solving them is the overarching habit of mind in the Common Core Math Practices.   Last year I witnessed students in both primary and secondary grades who could not persevere.  They believed that they could not do the math without teacher help.

The teachers and I worked together over several months to influence their mindsets.  Our strategy was three-pronged:

1. Use read-alouds and class discussions about these stories to help children start to build awareness that we can all learn and that making mistakes is a big part of learning.
2. Use ongoing think-alouds when modeling math to help students see what it looks like to be stuck.  Show them how to pause and think about what you know or what tool you could use to help you.
3. Create an anchor chart with the students, highlighting the tools and moves they can use to help themselves when they feel stuck.  

In the primary grades, students quickly connect with these stories about the importance of determination:

Try and Stick with It by Cheri J. Meiners, M. Ed.

It's hard to try something new.  When things get tough, the characters take us on a journey filled with flexibility, perseverance, and the importance of sticking with it.

I Can Do It!:  A First Look at Not Giving Up by Pat Thomas

The message in this book is about trying your best when you attempt something new and not being afraid to make mistakes.

If Only I Could! by David and Mutiya Vision

The child in this story gets frustrated when things don't go her way. Surprisingly, her baby sister teaches her the importance of perseverance.





In the secondary grades, these titles foster powerful discussions around perseverance:

The Most Magnificent Thing by Ashley Spires

A girl sets out to make something magnificent.  When things do not go as she hopes, she becomes discouraged and quits.  Her dog and faithful friend helps her to realize the importance of perseverance.

The Girl Who Never Made Mistakes by Mark Pett

Beatrice Bottomwell never makes mistakes.  When she ends up making a mistake for all to see, she learns that it's not a big deal to mess up.

Butterflies for Kiri by Cathryn Falwell

Kiri get frustrated that she cannot make origami butterflies.  She perseveres and through her creativity and determination, she learns more than just how to make butterflies.




When modeling in a second grade class, I let the students witness my struggle with adding coins.  Several students were quick to give me suggestions on what could help me - the coin chart, the one hundred chart, or the coin manipulatives.  At this point it was easier for them to offer someone else help because they did not personally "feel" the struggle.

Yet this discussion naturally led to charting the tools that can help us.

With the anchor chart in place, the students interacted with the chart to help them start to build their perseverance muscles.  They took ownership of their learning, and the confidence building was so evident.  Some students did continue to quickly come to the teacher and the teacher simply pointed to the anchor chart, redirecting the student.  

Yes, it is a process and each child moves through this process at his or her own rate, just as it's taken me a long time to start to embrace change.  The important thing to remember is that we are all growing.

Please share strategies you have found to foster positive thinking and perseverance with your students.  We all need to keep adding to our tool box.