“We All Have
Something
That Has
to Do with
Tens”:
Counting School Days, Decomposing
Number, and Determining Place Value
I
t is the seventieth day of school. As in many
other second grades, the children in this class
keep track of how many days they have been in
school. In the following vignette, the children share
number sentences that equal seventy. The second
graders and Mrs. K., their teacher, are gathered on
the rug in front of the whiteboard. Each child holds
a small blue mathematics journal, in which they
record their work. Keaton begins sharing expressions that equal seventy. As he dictates, Mrs. K.
writes “139 – 39 – 30” on the chalkboard.
Keaton: One hundred thirty-nine, then if you minus
thirty-nine, you’ll be back at one hundred. Um, and
then minus thirty. One hundred, then you go ninety,
eighty, seventy.
Mrs. K.: Oh, beautiful. You can even do that mentally.
Keaton: I did that.
By Anne M. Goodrow and Kasia Kidd
Anne M. Goodrow, agoodrow@verizon.net, teaches elementary mathematics methods courses at Rhode Island College in
Providence. She is interested in children’s mathematical thinking and preservice and in-service teacher development. Kasia
Kidd, kiddk@lincolnps.org, has taught for more than thirty
years, first as a special educator and currently as a secondgrade teacher at Lincoln Central Elementary in Lincoln, Rhode Island. She is interested in how
children’s oral explanations of their reasoning support the development of their thinking.
74
Mrs. K.: You did that. You did it mentally, and you
didn’t even use the number grid.
Winston shares next. He reads aloud how he has
made seventy, and Mrs. K. writes his expression on
the chalkboard: “1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +
10 + 20 + 10 – 10 – 1 – 1 – 1 – 1 – 1.”
Mrs. K.: Do you want to explain this one to us?
Winston: Well, my grandpa told me a trick. You
add the one with the nine, the eight and the two, the
three and the seven. [He draws a line connecting
the 1 and the 9 to make a 10.]
Mrs. K. [to the class]: Do you see how he made that
line there?
Mrs. K. [to Winston]: You see how you showed us
how the one and the nine go together and make a
ten? Could you do it for the other numbers you are
combining so we can see?
[Winston connects the 2 and 8, 3 and 7, and 4 and 6.]
Winston: It equals ten plus ten from here [pointing
to 1 and 9, 2 and 8. He then counts aloud.] “One,
two, three, four, five” [as he writes] “10 + 10 +
10 + 10 + 10” [under the long expression].
Winston: Then plus a five. So that equals five tens [plus
5]—equals fifty-five. And the fifty-five plus twenty
equals seventy-five. Seventy-five plus ten equals eightyfive. Minus ten equals seventy-five. Minus one, minus
Teaching Children Mathematics / September 2008
Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
one, minus one, minus one, minus one. Minus five
ones. Minus ten and minus five ones equals seventy.
Dr. A: What do people think about what Winston did?
Alex: He added one plus two, plus three, plus four,
but he knew he could make them all ten by adding
the 1 to the 9, the 2 to the 8, the 3 to the 7, the 4 to the
6. Then [he added] the 5 is by itself. Then you still
have the 10. And that’s ten, twenty, thirty, forty, fifty,
fifty-five … plus twenty equals seventy-five, plus ten
equals eighty-five, minus ten equals seventy-five,
minus one equals seventy-four, minus one equals
seventy-three, minus one equals seventy-two, minus
one equals seventy-one, minus one equals seventy.
Dr. A: Neat. So it really is helpful to know how to
make ten, isn’t it?
Mrs. K.: That making ten idea. Look at this. That
kind of reminds me of Keaton’s because he told me
that 139 minus 39 would give me what?
Children: One hundred.
Mrs. K.: Now I’ve got one hundred minus thirty
equals seventy. Does anybody see a ten in there?
Gianna?
Gianna: Thirty and seventy and a hundred.
Mrs. K.: And one hundred, right? So, that thirty and
seventy makes …?
Gianna: One hundred.
Mrs. K.: So, three tens and seven tens make …?
Gianna: One hundred.
Mrs. K.: And that’s the same as? Ten?
Gianna: Tens.
Mrs. K.: Ten tens. Exactly right.
A few minutes later, Maggie takes her turn sharing: “One hundred minus ten, minus twenty.”
“Minus twenty,” Mrs. K. repeats as she writes on
the chalkboard.
Mrs. K.: You want to tell us about that one?
Maggie: I knew one hundred minus thirty equals
seventy, but instead of using the thirty, I [changed]
it to one hundred minus ten. I knew twenty plus ten
equals thirty. So I did the minus ten, and then I did
the minus twenty.
Mrs. K.: Nice going. You know what Keaton said
as soon as you went up there to talk about it? What
did you say?
Keaton: I said, “It’s just like mine.”
Mrs. K.: He said, “It’s just like mine.”
Maggie: All of us who shared have something that’s
the same—Keaton, Winston, Jailene, me—we all
have something that has to do with tens.
Mrs. K.: Exactly. You do. You all have something
that has to do with tens. That’s great.
Teaching Children Mathematics / September 2008
Counting School Days
Counting the days of school is a popular routine in
many elementary mathematics classes. It is often
used only as a counting and grouping activity, and
on the hundredth day of school, the children bring
collections of one hundred objects to school. The
intention is to help them develop an understanding
of place value. Mrs. K.’s classroom is no exception
in this regard, but it is an exception when her students break apart the day’s number to create their
own equivalent numerical expressions.
What do the children learn about place value by
decomposing numbers in this activity? What does
their teacher learn about the children’s mathematical thinking when they share? How does Mrs.
K. respond to their sharing? How does she help
her students move forward in their mathematical
thinking? This article looks at the events that took
place on day 70 of the school year to illustrate how
decomposing numbers can help children construct
their understanding of place value.
Thus far in their sharing, we learned that
Keaton can count back by tens from one hundred
with ease, although we do not know if he can do
so from a number that does not end in zero. We
saw that Winston is very interested in numbers and
how they combine. He and other children know
pairs of numbers that equal ten, and when Winston
shares and Mrs. K. responds, the students see
that making tens can be helpful when combining
a string of single digits. Mrs. K. also pointed out
to the children that just as three ones plus seven
ones equal ten ones, three tens plus seven tens
equal ten tens. She helps the children extend
what they know about making ten by adding
ones to making one hundred by combining ten
tens.
Tyler shares
As the class continues, place value and ways to
make ten remain the focus of the discussion.
Tyler: Fifty-seven plus thirteen.
Mrs. K.: This is one I’d like you guys to talk to
each other [about]—you’re going to talk to your
partner. Now listen carefully. I look at the number
seventy, and I look at that, and I think “Oh, I need
to have seven tens there.” Now look at what Tyler’s
got. He’s got a fifty, so that’s five tens, and then
he’s got a thirteen, so I see another ten there. So
that’s five, six tens. So I’m just wondering, where’s
that other ten? Why don’t you talk to your partners
about that?
75
A buzz of children’s voices fills the room. One
girl says to her partner, “Fifty plus a ten would
equal sixty. And then seven plus a three will equal
another ten. That would equal seventy.” Here Mrs.
K. has used a “turn and talk” technique to involve
everyone in the mathematics; she wants everyone to
be thinking about ten. She makes explicit that fifty
has five tens and thirteen has one ten, although, in
retrospect, she might have first asked the children,
“How many tens does fifty have?” or “What does
the five in fifty mean?” before telling them that
fifty has five tens. Nonetheless, highlighting the
fact that seventy has seven tens, fifty has five tens,
and thirteen has one ten and asking the children
to find the remaining ten communicates to them
the importance of tens in our number system.
Mrs. K.: All right. Who would like to, who
would like to explain that to us? Liza, what’s
your thought?
Liza: Fifty plus the ten equals sixty, and the
seven plus the three is another ten. And that
equals seventy.
Getting “unstuck”
As the discussion continues, Tyler, the boy who wrote
“57 + 13,” encounters difficulty explaining how he
knows that fifty-seven plus thirteen equals seventy.
Mrs. K. gently supports and guides him through his
explanation, helping Tyler to clarify his thinking and
at the same time reinforcing mathematical concepts.
Tyler: You know, like the five, the fifty-seven and
the thirteen, so I added the thirteen, and then I
added up seven, so then I went [he stops talking
and is quiet].
Mrs. K.: So you had this [she writes “57 + 13”],
and you wanted to check and see if it really was
seventy?
Tyler: Yeah.
Mrs. K.: So how did you start?
Tyler: I started with the fifty-seven.
Mrs. K.: OK.
Tyler: So then I went up—fif—no, I went up thirteen again, and then …
Mrs. K.: You went up thirteen. What do you mean
by that?
Tyler: I mean that when I was at fifty-seven, I went
up thirteen.
Mrs. K.: Were you using a number grid [hundreds
chart]?
Tyler: Yeah.
Mrs. K.: Go on.
76
Tyler: So then I went up thirteen again, and I
thought to myself, that didn’t equal seventy. So I
went back thirteen.
Mrs. K.: I’m a little confused. So you’re saying
fifty-seven plus thirteen is not equal to seventy, or it
is equal to seventy?
Tyler: It is.
Mrs. K.: It is. And how do you know for sure?
Tyler: Because thir—no.
Mrs. K.: You want to start with your fifty-seven?
Tyler: Yeah.
Mrs. K.: Okay. So I’m at fifty-seven, and what
would you like to add to it first?
Tyler: I added a, a ten.
Mrs. K.: All right. So fifty-seven plus ten got you
where?
Tyler: Got to … [pause]… Got to sixty-seven.
Mrs. K.: Did this help? [She has drawn “57” and
“67” in boxes, vertically, as they are on the hundreds grid and has written “57 + 10 = 67.”]
Tyler: Yeah.
Mrs. K.: Now he’s at sixty-seven. So now what are
you going to do?
Tyler: So then I plus. [Tyler is silent.]
Mrs. K.: Where did this ten come from?
Tyler: It came from the fifty-sss—from the thirteen.
[Mrs. K. points to the 1 in 13.]
Mrs. K.: So this one stands for your ten, right? So,
what have you got left?
Tyler: Three.
Mrs. K.: Three. And sixty-seven plus three more
makes …?
Tyler: Equals seventy.
Mrs. K.: Nice job.
As we examine Mrs. K.’s words and actions, we
note that first she asked Tyler for clarification: “You
went up thirteen. What do you mean by that?” She
then asked about the tool he may have used. Moving “up” thirteen suggests using a number line or
number grid (hundreds chart). When Tyler was confused, Mrs. K. suggested that he start at fifty-seven.
Tyler chose to add ten but could not go any further
until Mrs. K. drew a visual—two boxes arranged
vertically and labeled “57” and “67.” After Tyler
said, “Sixty-seven,” she wrote, “57 + 10 = 67.” In
the discussion about where the ten came from, Mrs.
K. reinforced place value. The ten is part of the
thirteen, and now Tyler knows he has three left to
add to sixty-seven. Mrs. K’s scaffolding has helped
Tyler, and very likely other children, to think about
place value (the meaning of the one in thirteen) as
well as how to add one ten to a number.
Teaching Children Mathematics / September 2008
Adding tens and adding to
make ten
The class continues as children share their ways to
make seventy and Mrs. K. asks for one more child
to explain where the seven tens are in 57 + 13.
Mrs. K.: Is there one more person who can tell me
where the seven tens are? Okay, let’s see. G
­ abriella.
Gabriella: Um … um … I took away the three and
the seven for a while. And then …
Mrs. K.: So you ended up with a fifty and a ten.
Gabriella: Umhmm, and I knew that a fifty and a
ten would equal sixty.
Mrs. K.: Umhmm.
Gabriella: Then plus three plus seven would equal,
would equal ten. So fifty plus ten will equal sixty
plus a ten will equal, will equal seventy.
Mrs. K: Nice going. So you broke that fifty-seven and
that thirteen into fifty, ten, three, and seven. Nice job.
Dr. A: So she knew how to add tens. She knew that
seven and three made ten.
Mrs. K: That’s handy.
Child: We have it on our desks from you. That’s
why me and Jailene went to get it.
Mrs. K: Oh, because it’s …
Child: Ways to make ten.
Gabriella: Equals ten.
Mrs. K.: And that’s pretty important …. It came in
useful when …
Gabriella: Winston.
Mrs. K.: Winston. When Keaten needed seventy;
one hundred minus thirty.
Child: One hundred thirty-nine minus thirty-nine.
Mrs. K.: He had that one hundred minus thirty right
in there.
Dr. A: And when Winston had added all those
numbers.
Mrs. K.: And when Winston added all those numbers too. Nice job.
[Economopoulos et al. 1998, p. 40] and “The Name
Collection Box” in Everyday Mathematics [Bell
et al. 2002]), students have many opportunities to
look at how numbers are composed, and, in doing
so, they develop their understanding of the structure
Figure 1
Examples of children’s expressions for 18 are from early in the school
year.
(a) Mrs. K. used the mathematical terms expanded notation and commutative property. She inserted parentheses to show how one student added fives
(Kidd 2005).
10 + 8 = 18 expanded notation [the teacher added the label]
18 + 0 = 18 0 + 18 = 18 cmoohivpopde (commutative property)
19 – 1 = 18
13 + 5 = 18
42 – 24 = 18
(2 + 3) + (2 + 3) + (2 + 3) + 3 = 18 [the teacher added the parentheses]
17 + 1 = 18
100 + 100 –100 + 100 – 80 – 20 – 3 – 7 – 3 – 7– 3 – 7– 3 – 7 – 3 – 7– 3 – 7– 3 –
7 – 3 –7 – 3 – 7 + 8 – 8 + 8 – 8 + 4 + 4
(b) The commutative property makes sense, and children remember the term
when they have used it in their work.
Gabriella, like Liza, sees that groups of ones,
in this case seven and three, can be added to make
ten. However, these children treat this sum of ten as
another group of ten. As Liza put it, “Fifty plus the ten
equals sixty, and the seven plus the three is another
ten.” Their discussion suggests that they see that ten
can be ten ones or one group of ten, an important
developmental step in understanding place value.
Decomposing the Number
of the Day
Through this classroom routine (entitled “Today’s
Number” in Mathematical Thinking at Grade 2
Teaching Children Mathematics / September 2008
77
of our number system as well as explore arithmetic
operations.
As the children break apart numbers, they work
with part-whole relationships. Part-whole relationships are often considered in the context of number concepts and conservation of number, as, for
instance, the different combinations that make a
certain quantity (e.g., 5 can be 2 and 3, or 1 and 4,
etc.). Our view of part-whole relationships includes
place value. Place value is more than identifying
the ones, tens, and hundreds columns. Understanding groups of ten grows out of children’s own logic
as they work with numbers in meaningful activities
Figure 2
The constraint of using three addends
was given on day 94. This student first
decomposed the 4 and then the 90.
Figure 3
Here are five examples from day 108, when students were asked to
write expressions using subtraction in this format: ___ – ___ = 108. Some
children decomposed larger numbers, and others wrote story problems.
108
8,000 – 7,892
300 – 192
708 – 600
Heather has 109 dollars. She loses 4 quarters. How much money does she
have now?
Liza knows 110 Russian words. She forgets 2. How many Russian words
does she know?
78
such as decomposing numbers to make expressions
equivalent to the number of the day. Over time and
with many numerical experiences, children come
to understand the relationship between numbers
and groups of tens and ones (Burns 1993; Kamii
1985, 1989; Ross 1989), and the idea that a digit
can simultaneously represent a number of tens or a
number of ones (e.g., the 5 in 52 can represent five
tens or fifty ones).
How the activity evolves
A typical mathematics class, like day 70, begins
with children working individually in their mathematics notebooks to create mathematical expressions equivalent to a particular number. As the
children work, we circulate, ask questions, and
offer help if needed. Our goal during this time
is to encourage the children’s own mathematical
thinking. Questions can be general, such as, “How
did you think about that?” and “How does your
pattern work?” or more specific, such as, “What
will happen if you add another ten?” and “Is there
something you know about [another number] that
you can use to help you?”
Next, the class comes together, and the children
share and discuss their numerical expressions. Discussion is a critical component of the activity; children’s ideas are made public, and mathematics is
made explicit, as we saw in the vignette. We focus
the children’s attention on mathematical ideas that
otherwise may remain embedded within the children’s work. Although we have topics in mind, such
as place value, topics may also come from the children themselves. Discussion topics have included
the inverse relationship of addition and subtraction,
negative numbers, and, of course, place value.
Early in the year, few constraints are imposed on
the task. Very long expressions, such as the last one
in figure 1a, capture children’s interest; however,
children have to learn to keep track of their computation. They learn about chunking numbers to
subtract and about making zero; in one memorable
class, the children come to the conclusion that if
you subtract the same amount as you add (or vice
versa), the result is zero. Opportunities arise for
using mathematical language in context. Children
remember the commutative property, and it makes
sense to them, when it is connected to their own
work (see fig. 1b). As the year continues, Mrs. K.
provides mathematical constraints and challenges,
and the children’s work increases in sophistication
as their knowledge of our number system grows
(see figs. 2 and 3).
Teaching Children Mathematics / September 2008
We believe that children’s continued experiences
decomposing number as they keep track of their
school days encourages development of understanding place value as well as practice using the
number system. Such experiences may very well
provide a much-needed foundation for children’s
work with algorithms for addition and subtraction;
these algorithms can be viewed as particular ways of
decomposing to combine or separate numbers based
on place value. Such experiences provide additional
rewards, too. They support children in the quest to
learn mathematics meaningfully and well, in taking
risks, and in learning from one another.
References
Bell, Max, Jean Bell, John Bretzlauf, Amy Dillard, Robert Hartfield, Andy Isaacs, James McBride, Kathleen
Pitvorec, and Peter Saecker. Everyday Mathematics.
2nd ed. Chicago: Wright Group/McGraw Hill, 2002.
Burns, Marilyn. Mathematics: Assessing Understand-
Teaching Children Mathematics / September 2008
ing, Part 1. White Plains, NY:
Cuisenaire Co. of America. 1993.
Video.
Kamii, Constance. Young Children
Reinvent Arithmetic: Implications
of Piaget’s Theory. New York:
Teachers College Press, 1985.
———. Young Children Continue to
Reinvent Arithmetic (2nd Grade): Implications
of Piaget’s Theory. New York: Teachers College
Press, 1989.
Kidd, Kasia. “A New Routine for Name Collection Boxes.” TeacherLink 14, no. 1 (Fall 2005):
16–17.
Ross, Sharon. “Parts, Wholes, and Place
Value: A Developmental View.” Arithmetic Teacher 36 (February 1989):
47–51.
Economopoulos, Karen, Joan Akers, Doug
Clements, Anne Goodrow, Jerrie Moffet,
and Julie Sarama, eds. Mathematical Thinking at Grade 2. In Investigations in Number,
Data, and Space series. Menlo Park, CA:
Dale Seymour Publications, 1998. s
79