Before CCSS, we used to teach students a standard algorithm by showing them the steps on how to borrow or regroup without much explanation than " we need to borrow a group of tens" language. I know in the past in my own teaching, I didn't say much about place value or talk about why we could exchange and regroup. So, being committed to mastery of this standard and Math Practice Standards 1 & 2, I wanted to really get at the heart of their thinking by asking them: What does it mean to regroup?
The discussion began with great confidence as hands shot up in the air. But, they started to tell me how. Even as I persisted to coach them to tell me why, it turned into how. I went back and emphasized the word "mean." It still didn't generate the answers I had hoped for. And yet, it gave me a clear picture of where I needed to start.
One boy simply said "It [regrouping or borrowing] keeps it [subtracting] from being all messy." And there is some truth in that! I never thought about it like that! Regrouping in the methods of crossing numbers out does help organized the algorithm. He sees this as a process of organization and that it also a good thing. It is a place to start.
Warm up: I opened the lesson by asking them to solve a regrouping problem involving a number in the ten thousands and a number in the thousands that I had written on the board.
They wrote the equation in their math journals and began to solve. I visited each student looking for how they lined up their numbers, how they regrouped and how they showed that regrouping,if they regrouped the right number, and if they remembered to place commas correctly. This quick assessment told me where they were in their understanding of regrouping. I saw lots of eyeopening things! I even saw this: Whoops!
Teacher Think Aloud: I used the problem I had written on the board for the warm up, as the model for my "talk through."
I told my students that I was going to regroup by using place value in my language to show them how I can think through exactly what I am doing. At this point in their learning, I am certain they have mastered knowing the place value names through one million.
I invited them to join me in my "talk" as soon as they thought they could.I started reading from the ones column, using place value words and regrouping as I added. Several joined in, but not all of my students did. I could see furrowed brows and a level of discomfort in even trying. They had never talked like this about their adding!
I am modeling MP7 (Making use of structure) and extending the ideas of 4.NBT.A.1. I truly believe this rigorous practice of reading numbers aloud as they add using place value in their language, will help them become more aware of place value, and much more fluent in regrouping in standard algorithms.
The second time around, I instructed them to say each number in each place value along with me. I slowed way down and watched to see that everyone joined in: 6 ones + 8 ones = 14 ones. The ten is regrouped to the tens place...1 tens + 5 tens + 2 tens = 8 tens, etc. I saw that they were on the edge of their seat wanting to keep up. All were engaged. All were on task. I kept it light and fun with my voice, pauses and tempo. I heard more and more students join in, and everyone was trying.
This continued until each place value was added and regrouped.
This models for them how their thinking should be and supports the steps to fluently adding a standard algorithm. (4.NBT.B.4) Proficient math students can model their thinking about addition and regrouping through this process as expected in MP4.
It's hard! It's really hard. I think it is tough because they were so used to just going through the motions. Talking it through feels a bit awkward. It's risky to do it out loud too!
It sounds like this...Talking It Through
After my "talk through" was done, I wrote a few multi-digit algorithms on the board for them to practice together. My students use the Educreations ap for note taking or solving problems. Educreations is an ap that is wonderful because they HAVE to record a voice-over in order for it to save properly. They talk through their addition and have it all recorded to listen to themselves or to share. See Educreations.com
Otherwise, pencil and paper is just fine!
They need to practice until they can show they are fluent or at least nearly fluent in using place value language to regroup.
I closed my lesson with a discussion about what they had learned from the experience and asked if they were thinking differently about regrouping now. I asked them, "How is your understanding of the meaning of regrouping different now than it was at the beginning of class?"
One of my highest level students raised her hand and told me that it felt weird to think out loud and hear her voice talk about regrouping because she had never thought about it as she was adding before. I think that this comment helped me see that the lesson was successful, but I also know that it will take my constant coaching to not forget to talk it through as a support for thinking.
One website I like to use for practice at home is IXL. I use this site for many skills. It can be set for CCSS or state standards. I decided to use this as an assignment for fluency in adding and for practice in talking through addition with larger numbers.
I clicked on 4th grade or Level F: I picked additio. I then assigned lesson B.1 for your middle students and B.2 for higher end students. I can differentiate easily with this website.
I told them that I wanted them to write their problems on paper, solve and put the numbers in the answer box after they solved it. I also would be collecting all of their work. I told them to work 20 minutes or do at least 10 problems. Otherwise, they will work 20 minutes and just do a few because the timer will just keep running until it realizes there is not any activity. I also assigned that they should "talk through" their work as they solve.
To ensure that they practice aloud, I told them that they will choose one problem to share and recite on the smart board in class tomorrow. That way, I know they have practiced at least one orally.