Kate Coombs

Education with Imagination

Mathematics

One of the keys to understanding the world of math is the concept of place value—which organizes our number system. Yet too many children are fuzzy on this admittedly challenging idea, and that fuzziness blurs their math work in related areas. An important task affected by the understanding of place value is regrouping (or carrying and borrowing) in multi-digit addition and subtraction.

Please understand: the method or algorithm we teach in schools for multi-digit addition and subtraction is a compressed version of what really happens. It's a shortcut! Naturally, when we teach kids a shortcut as if it were a "longcut," they don't tend to know what's going on.

It is not surprising that we've used compressed versions of written computation for centuries, since they promote speed. However, in this day of cheap calculators, speed can and should take a second place to comprehension. In an ideal world, I would recommend first teaching addition and subtraction in expanded form and later introducing those speedy shortcut algorithms. Of course, that's not how it's being done in schools today. Therefore, the following method is presented as a remediation tool (for about 3rd-5th grades). In using it, you will need to tell kids you're going to show them another way to add and subtract simply to help them understand the school method. Later in the process, you should have students compare the two methods by doing the same problems both ways.

Numbers as Nicknames

We must begin by teaching students a simple fact: the very numbers we use constitute a shortcut. This is a concept I try hard to instill in children as a precursor to dissecting the algorithm. I use a fairly commonplace analogy to do it. I ask one of my students her full name. Let's say it's a nice long one: Elizabeth Ann Scarlet-Jones. Then I do a little skit, like this:

Well, good morning, Elizabeth Ann Scarlet-Jones. How are you today, Elizabeth Ann Scarlet-Jones? Did you do your homework, Elizabeth Ann Scarlet-Jones? What's your favorite pizza topping, Elizabeth Ann Scarlet-Jones? Elizabeth Ann Scarlet-Jones, your red bow is very pretty!

Next I ask, "What's funny about the way I'm talking?" The class agrees that no one uses all those names constantly in conversation. What do we call this young lady most of the time? Just Liz. And that is what we refer to as a nickname.

Finally I impart the secret: most of the numbers we use every day are also nicknames—they're short versions of something longer. Here's an example of what we discuss:

Number Nicknames

13

25

38

147

209

Number Names

10 + 3

20 + 5

30 + 8

100 + 40 + 7

200 + 9

The class then practices converting numbers from short to long form and back until they're pretty comfortable going either way. After a while, we make sure to take a good close look at zero's role. For example, what happens if we don't use it to show that there aren't any tens in 209? We end up with 29, a different number.

Of course, you may be wondering how to address the fact that numbers are bundled into ones, tens, and hundreds, etc. One thing we can do is simply work off of students' existing knowledge of how to say the numbers, since most of the number names contain the breakdown: e.g., twenty-three is easily converted to 20 + 3, or three hundred eighty-nine to 300 + 80 + 9. Teach kids to say the numbers and listen for those place value pieces.

The exceptions to this spoken-name rule are found in the numbers 0-10 and 100 and 1000, etc., which don't need breaking down, and in the numbers 11-19, which are compacted even when pronounced. Help kids find the place value numbers hidden in the words eleven, twelve, thirteen, and so on up to nineteen.

You can also try having kids do an exercise in which they dissect the numbers and then add the parts in vertical columns, like this:

Here's an example that shows how 0 works:

Have kids do that one over, showing that the 0 isn't necessary. However, we can include it, and we'll use it to help us out in the expanded form method demonstrated below.

Many teachers have ways of talking about converting numbers to expanded form, but this "nickname" approach tends to make a lot of sense to kids. What's most important about it is explaining that the counting numbers are not the "true" form; they are an abbreviation of the true form. It's also vital to spend enough time on these concepts so that they stick, and so that kids feel very comfortable with the conversions in both directions. (School math books tend to rush through skills and hurry on to the next set of tasks.)

Expanded Form Addition

Once students have thoroughly practiced converting numbers from short to long form and back, they are ready to try adding in expanded form. I have found that teachers are sometimes uncomfortable with the fact that my addition method involves regrouping at the bottom of the problem. This doesn't phase kids nearly as much as it does adults, however. You can simply explain, "Okay, at school you do your regrouping at the top of the problem, but in the way of adding I'm going to show you, we regroup at the bottom. Here's how...." Remember: you can throw a lot at kids, and they'll flex with it!

Begin with two-digit problems that do not require regrouping. The only really tricky thing is that when adding in expanded form, we have two kinds of + signs—the small "glue" sign that holds the expanded form pieces together and the big "operation" sign that represents addition. Point this out to kids and have them use different sizes of + signs. They should catch on fairly quickly. Sometimes I call the big operation sign "the big action."

Here's what 15 + 34 looks like in expanded form:

This may seem like a lot of extra work, but as I told a student who said as much, "Yes, but I'm going to assign you fewer homework problems." He liked that! (Remember, the goal is to understand what happens when we add or subtract using the abbreviated school algorithms.) We might also explain to students that we are going to take the problems apart and then put them back together so that we can see what makes them tick—the same way we might take an old watch or small appliance apart and examine it.

What about a problem with 0 as a placeholder? Here's an example:

You could get fancy with 0, 00, and 000, but I wouldn't recommend it, since it's mathematically confusing, not to mention incorrect. In contrast, 20 + 0 really does equal 20!

Have kids practice on two-digit problems for a few days before moving on to three-digit problems such as the following:

Regrouping

Now let's take a look at regrouping. Basically, I have the students write the sums in expanded form below, accumulating off to the left. Then I ask them to group the sum numbers by circling like place values. You'll see that once we get below the equation line, the sum numbers no longer line up with the problem numbers. That's okay—really! Just tell students that this will happen: "You'll find that sum numbers will move off to the left and will probably stop lining up with the top." (If this drives you nuts, you can leave more space so that the lining up will still work.)

The most important task is converting sum numbers to expanded form as you go. Remind kids to do this, and help them practice! Just tell them that the sum numbers will all be written in long (or expanded) form. For example, in the problem below, 6 + 6 will be recorded below the equation line as 10 + 2. This will take a little getting used to at first, but they'll get the hang of it. You can start by having them do these conversions orally as part of the addition process.

Note: A hidden benefit to this work is that certain aspects of algebra will be easier to grasp when they are encountered in a few years. That is, students will be more comfortable with breaking numbers apart for distribution and seeing them in "broken form" on either side of an equation.

Have kids practice on two-digit problems for a few days before moving on to three-digit problems such as the following. Note that students may need a review in order to convert sums such as 120 (from 70 + 50) to expanded form as 100 + 20.

Bump-Ups

The only other twist that can occur is what I call "bumping up." A bump-up is when the regrouping results in a number that belongs to a higher place value. We can indicate bump-ups with arrows underneath, as in the following example:

After some practice, kids will be able to skip some of these final steps and go straight from 1000 + 400 + 90 + 10 to the sum, 1500. (Notice that this problem ends up having no tens or ones, but the place value zeroes appear in the number 500.)

Have students look at other bump-up problems and compare them to figure out what creates a bump-up situation. (Hint: it has to do with sums of 9!) Here's another one. I call this a double-bump!

Comparison with the Traditional Algorithm

Once students are comfortable using expanded form addition, have them work problems using both this method and the traditional school method and then compare what's happening in the paired problems. We can essentially use expanded form addition to translate the abbreviated school algorithm.

Expanded Form Subtraction

You'll find that expanded form subtraction looks a lot like the traditional algorithm. The one bone of contention I've encountered is when to regroup. I've sat with groups of teachers who have debated madly about whether the borrowing should be done one place value at a time or should be completed all at once, before beginning the subtraction itself. Clearly, the procedure has been taught both ways in different regions and classrooms. The truth is, subtraction regrouping works just fine either way. So, for better or worse, I will be using the latter approach here. As I tell the kids, you could do it the other way, but why try to borrow from somebody who's short on cash when you can go straight to the rich guy? This does, of course, require students to analyze the problem in its entirety early on, not such a bad step to follow!

In doing expanded form subtraction, students will again use those little "glue"-type addition signs within any expanded form numbers, but a large minus sign for the (vertical) subtraction operation. However, unlike in expanded form addition, you'll find that the numbers in the expanded form differences will line up below the equation line, since the regrouping takes place at the top of the problem.

Here's a two-digit example without regrouping:

This is a three-digit example without regrouping:

Regrouping

Notice how this procedure "spells out" what's happening in the traditional algorithm. Let's start with a two-digit example:

The student will need to use his expanded form conversions again, in this case to subtract 4 from 10 + 2, or 4 from 12. When students hand-write this, it's easier to show that the 10 goes with the 2, or to indicate any additional borrowing, as shown below! Here I've indicated the combinations with boxes for the sake of clarity, something you may want your students to do.

Here's a three-digit example with regrouping.

And one more example:

Comparison with the Traditional Algorithm

Again, once students are comfortable using expanded form subtraction, have them work problems using both this method and the traditional school method and then compare what's happening in the paired problems. We are using expanded form subtraction to translate the abbreviated school algorithm.

In Conclusion

When I demonstrated expanded form subtraction to a group of long-time teachers, one teacher said, "But you're putting a hundred in the tens column!" I took a deep breath and said, "Yes, that's exactly what we're doing, and we want the kids to understand that." In this case, of course, the teacher had learned the shortcut algorithm herself in elementary school and had never been completely clear on what was really happening. Which is all the more reason to teach children what's going on inside the algorithms while they're still young enough for place value knowledge to enrich their total math practice.

Now, somebody's probably kvetching about my methods revisiting the much-maligned turf of the so-called "new math." I personally didn't learn the new math, though I know people who did and wish they hadn't. I will tell you that what I'm talking about isn't deep math theory—it's nuts and bolts. What's being called "mental math" is slowly making its way into today's math books, and most of the time it has to do with various ways of taking numbers apart and putting them back together, often in relationship to place value. Why? Because researchers have observed that these activities are practiced by kids who are really good at math, kids who understand just what makes numbers tick. The child who's comfortable taking numbers apart and putting them back together has a better chance of succeeding in math class—and in navigating her way through the amazing world of numbers.