# Teaching Math for a Deep Understanding

It’s no surprise that subtraction with regrouping is a difficult concept for kiddos. If you were asked why it’s difficult, what would you say? In her groundbreaking book, Knowing and Teaching Elementary Mathematics, Liping Ma devotes a whole chapter to a comparison of how American and Chinese teachers approach instruction of subtraction with regrouping. Bottom line, American teachers typically teach the standard algorithm, often still referring to the process as “borrowing”, while Chinese teachers draw on a deeper understanding of decomposing numbers, teaching math for a deep understanding of the meaning behind the process.

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So, how do we teach for that deeper understanding? In Kathy Richardson’s book, How Children Learn Number Concepts, she makes the point many times throughout the book that kids can often appear to understand more than they really do. We have to probe deeper to find out what they really know. When I’m working with a student on regrouping, my favorite question is to ask if the number is still the same after regrouping. ย For example, say we are subtracting 81 – 27. The student will proudly announce that they need to trade a ten to get more ones. They show the work by crossing out the 8 tens, leaving 7 tens, and adding 10 more ones to the original 1 one, giving 11 ones. No subtraction has taken place yet–just the regrouping. Then I hit them with my question, “Do you still have 81?” Nine times out of ten the almost immediate answer is no. Really?, ย I ask, Let’s count. So that’s what we do and much to their surprise, they find out that 7 tens and 11 ones is still 81. You would seriously think it was a magic trick they are so surprised.

Sooooo, what piece of understanding is missing? Think about how we traditionally teach place value. We use manipulatives, such as base ten blocks, but we typically only represent numbers one wayโ35 is usually only shown as 3 tens and 5 ones. So here’s what you do. ย Have students work in pairs to show you the number 35 with base ten materials. Of course, they will show 3 tens and 5 ones. Then ask them to show you another way (I love the look of total confusion on their faces the first time you do this). No more hints, just let them think about it. I can almost guarantee that at least one pair will trade a ten for 10 ones and show it as 2 tens and 15 ones, at which point you ask them to explain it. Then ask for another way, and most pairs will probably show 35 as 1 ten and 25 ones. It helps to have them document their work using expanded form: 30 + 5 = 25 + 10.

Along these lines, I made a couple of freebies to use in your workstations AFTER kids have had concrete practice with the concept. The first features 2-digit numbers and pictorial representations using base ten blocks. Each number has four cards. You can use the cards for a matching activity, but there are other ways to use them as well.

• Put only the numeral cards in a workstation. Students use base ten blocks to build the number two different ways and draw sketches in their math journals.
• Use the cards for a game of War. Hand out the cards evenly between players. Each player turns over a card. The player with the greatest number takes both cards.
• Put the pictorial cards in a workstation. Students take a pictorial card and write the numeral, word form, and expanded form for the number shown.

There is also a set with 3-digit numbers. Notice that these cards are completely abstractโthat is, there is no pictorial representation. A suggestion would be to first use the numeral cards only with base ten blocks and have students represent the numbers two different ways. Remember, concrete experiences build understanding. Then you can use the cards for matching or in any of the ways outlined for the 2-digit cards.

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1. Anonymous says:

This is great! I love your page!

1. Donna Boucher says:

Thank you so much!

2. Thank you, Donna! Knowing and Teaching Elementary Mathematics is now on my summer reading list.

3. AmberO says:

These Freebies are great Donna! Love the enlightenment for today; lightbulb moments happen for adults, too:) Never thought about how we teach regrouping as a process to be memorized rather than ensuring children actually fully grasp how and why they’re doing the process. Have a wonderful Independence Day!

1. Donna Boucher says:

Glad it was enlightening, Amber. It’s interesting how just small shifts in what we do can produce amazing results! Enjoy your 4th. ๐

1. Donna Boucher says:

Absolutely! It should just be our standard way of teaching place value.

4. hj says:

I am so glad to see you reference Ma’s book! It changed my way of teaching when I first read it and I recommend it to everyone!

5. starsie says:

This is a fantastic resource, thank you for sharing! I’m returning to the classroom after two years spent as an instructional coach (K-5 math) and one of the big frustrations was watching children get pushed into using a symbolic rote algorithm that they really didn’t understand, and teachers feeling they “didn’t have time” to spend using manipulatives and building the concept of composing/decomposing. ๐

1. Donna Boucher says:

Amen and hallelujah!

6. Anonymous says:

Thank you for explaining it. I’m a mom to a rising 2nd grader who has a penchant for numbers, & had been regrouping w/addition since LAST summer,was trying to figure out regrouping w/subtraction last October. I shared the way I learned after checking w/the gifted teacher. I just called my son over to ask him the question you posed above about the equation and he did not realize the top number still was 81!
Will definitely work on a deeper understanding, he is pretty good at basic decomposing/composing, but not to the level you share. I know 2nd grade will explore it, but he is moving onto division & multiplication so I know he needs to understand numbers better NOW:)

Thanks again

1. Donna Boucher says:

So cool when you have a child handy to try out something new! I’m glad you found the post useful.

7. Anonymous says:

Thank you for sharing….another awesome resource is Number Talks….Van De Walle is one of my favs too! Happy Math!

1. Donna Boucher says:

Right! I use both regularly! ๐

1. Donna Boucher says:

Awesome, Stacy! Thanks so much for sharing. ๐

8. Jessy says:

I am a third grade teacher and still find it necessary to teach subtraction using base ten blocks. I tried asking the same question to my students and at first, they also said that the number changes after trading! Thanks for this post I was quite affirmed that how I do subtraction in my class gives better understanding of the concept. โบ๏ธ

9. Lori says:

Love these. Let’s keep on mathing!