@*#?
Photograph by Jo Ann Cady; all rights reserved
WHAT IS THE VALUE OF
Deepening Teachers’ Understanding of Place Value
By Theresa M. Hopkins and Jo Ann Cady
Theresa M. Hopkins, [email protected], is a postdoctoral fellow in mathematics education at the University of Tennessee,
Knoxville, TN 37996, with a background in middle and high
school mathematics. She is interested in creating quality
professional development experiences for beginning and inservice teachers and also in rural mathematics education. Jo
Ann Cady, [email protected], is an associate professor of mathematics education at the University of Tennessee, with a background in elementary and middle
school mathematics. She is interested in teachers’ beliefs, pedagogy content knowledge, and
assessment practices.
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A
s faculty members of the Mathematics Education Group in the College of Education,
Health, and Human Sciences at the University of Tennessee, we are responsible for instructing
both preservice and in-service teachers through
courses and professional development activities.
One topic we address is teaching place value to
elementary school students. Teachers’ familiarity
with the base-ten number system, however, can
prevent them from fully comprehending the diffiTeaching Children Mathematics / April 2007
Copyright © 2007 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.
culty these students have when trying to understand
the abstract concept of place value. This article
presents our evolving lesson in addressing this
difficulty.
The Significance of
Understanding Place Value
To circumvent our mathematics education students’
familiarity with the base-ten number system, we had,
in the past, used base-five blocks to investigate place
value. However, rather than working within the basefive system, many of our students simply tried to
convert from base-five to base-ten. Further, both our
preservice and in-service teachers continually read
numbers incorrectly—for example, they read 105 as
“10” rather than as “one zero base-five.” This realization led us to brainstorm ideas for activities that
would produce a cognitive dissonance in these teachers, forcing them to think about place-value concepts
such as these identified by Van de Walle (2007):
1. Sets of ten (and tens of tens) can be perceived
as single entities. These sets can then be
counted and used as a means of describing
quantities. For example, three sets of ten and
two singles is a base-ten method of describing 32 single objects. This is the major principle of base-ten numeration.
2. The positions of digits in numbers determine
what they represent—which size group they
count. This is the major principle of placevalue numeration.
3. There are patterns to the way that numbers
are formed….
4. The groupings of ones, tens, and hundreds
can be taken apart in different ways. For
example, 256 can be 1 hundred, 14 tens,
and 16 ones but also 250 and 6. Taking
numbers apart and recombining them in
flexible ways is a significant skill for computation…. (p. 187)
The Orpda Number System
Because of the difficulties students confront in
understanding place value, we as facilitators
decided that we would create a new number system—which we named Orpda—that would use
symbols rather than numerals to represent values.
This approach had the advantage of making the
activity more abstract so that the teachers’ experience with place value would be similar to that of
Teaching Children Mathematics / April 2007
their elementary school students’ introduction to
place value. In previous workshops and courses
that we had conducted, the teachers’ familiarity
with base-ten numerals often interfered with their
learning, but this approach intentionally placed
them into an entirely new number system. We also
wanted the teachers to explore the importance of
concrete models and grouping activities, so we
provided multilink cubes for early sessions and
base-five blocks for later sessions.
To begin the activity, we placed a blank transparency on an overhead projector, drew an empty circle
to create a group containing no objects, and told the
teachers that the symbol ~ represented the number
of objects inside the circle. The ensuing discussion
included our statement that ~ represented nothing.
Then sets containing 1, 2, 3, and 4 objects were
illustrated and represented by different symbols as
shown in table 1. To be sure that the teachers were
comfortable working with the Orpda symbols, we
asked them to indicate the symbol that represented
several different sets of objects ranging in number
from 0 to 4 objects.
Table 1
Number of Objects and Representative
Symbol in the Orpda Number System
1 object
~
*
2 objects
@
No objects
3 objects
#
4 objects
&
We explained that these five symbols were the
only symbols in the Orpda number system and then
challenged the teachers to use them to represent a
group having 5 objects. Anticipating that the teachers might need some time to discuss answers to this
question within their groups, we were surprised to
see several hands go up immediately, but we allowed
a few minutes for others to think about the question
before soliciting responses. Although we expected
the answer *~, representing a single group of 5 and
no units, several teachers gave us alternative answers
that quickly pushed all of us, teachers and facilitators
alike, out of our comfort zone.
The first teacher gave the answer *&. We wrote
her response on the board and asked her to explain
her answer. She stated that * represented 1 and
& represented 4; therefore, *& represented 5. We
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acknowledged her answer, recorded it on the board,
and then asked if anyone had a different one. A second teacher gave the answer @#. Again, we wrote
the answer on the board and asked for an explanation. The teacher replied that @ represented 2 and #
represented 3, for a sum of 5. These answers were
unexpected, but even more unexpected was that no
one in the group of nearly fifty teachers suggested
what we considered the correct answer.
Rather than immediately commenting on these
answers, we began a discussion about how to represent the value 6. This gave us time to think about a
new direction and further evaluate the participants’
mathematical thinking. Again, the teachers gave
several different answers, some using two symbols
and some using three, such as @@@. They discussed the validity of the various representations
and concluded that all were valid. We accepted these
justifications and acknowledged their reasonableness
but then posed this question: “If we have many ways
of representing the same number, how would we
know which to use?” We countered the explanation
that @# represented the value 5 with an example
from the base-ten system, pointing out that although
2 + 3 represents the value 5, the representation 23
does not. These arguments led to the realization that
unless we could create a unique method of representing values greater than 4, using the Orpda number
system would be very confusing.
Relating to the base-ten system
We asked the teachers to think about the numerical,
or place-value, relationships in the base-ten number
system and then create a unique symbolic representation of 5 in the Orpda number system. After
much discussion within their groups, the teachers
suggested the answer *~, which we displayed on the
board. Several teachers were skeptical and asked for
an explanation. The teacher who had suggested this
answer compared *~ to 10 in the base-ten system,
saying we had one group of 5 with no singles left
over; this formulation compares with one group of 10
with no singles in the base-ten system (see fig. 1).
Many of the teachers accepted this representation
and its explanation, but some did not. A discussion
regarding the use of ~ (zero) ensued. Several teachers suggested that ~ represented nothing, as defined
at the beginning of the lesson, and should not be
used. Others countered that ~ represented the idea
that there were no units but rather one group of 5
objects, hence the use of *~ to represent the value
displayed. They maintained that their idea was
related to the idea of 10 in the base-ten system being
represented by one group of 10 objects and no units
or ones. We then provided more information on the
invention of zero and its use as a place holder. The
teachers’ struggles with the concepts of zero, grouping, and the position of digits within a numeral parallel the struggles their elementary school students
might have with the same concepts.
Using manipulatives to
understand a number system
Using multilink cubes, the teachers were then asked
to count and represent sets of objects up to 30 within
the Orpda number system. We found it interesting
that the teachers used the multilink cubes to represent the number of objects but did not use them to
create groups of 5. As we continued to count, the
teachers began to see the patterns that developed,
similar to patterns that elementary school students
find when completing a hundreds table.
Several teachers connected the ideas from the
base-ten number system to the Orpda number system
and quickly created symbols to represent the values
of sets of objects through *~ ~ (25). Others needed
more time to think and some guidance from the facilitators. We let the teachers struggle with the problem
Figure 1
Combining units to create a rod in the base-ten number system (a) and the Orpda (base-five)
number system (b)
+
1a. 10 = 1 rod and 0 units
436
+
1b. *~ = * rods and ~ units
Teaching Children Mathematics / April 2007
on their own until we felt the frustration level rise;
only then did we step in to offer suggestions to those
who were frustrated to the point of no longer trying.
At this point, we suggested that they use the cubes to
model the values. Rather than show or tell the teachers what to do with the multilink cubes, we continued to ask questions such as these:
• “What do you already know about the symbols
for numbers in the Orpda system?”
• “How can you represent these symbols with
cubes?”
• “How would a set of X X X X X X objects be
represented in Orpda?”
• “What would happen if you added one more
object to the set? How would you represent this
value?”
• “What happens when you have X X X X X X X
X X X objects?”
• “What patterns do you see?”
After a short time, one teacher suggested that we
create a group of five rods, or groups of 5, to make
a square (flat) and use the symbol *~ ~ to represent
this value. Many of the teachers could now make the
connections between the base-ten system and the
Orpda system. They correctly represented the Orpda
symbol *~ ~ ~ as a cube, or 5 groups of 25 (5 flats).
At the conclusion of many of our workshops,
we encouraged a discussion about the participants’
frustration level. We were surprised by the teachers’
vehement opposition to using the word frustration to
describe feelings about mathematics. They viewed
this term as negative and stated that they did not use
it with their students during their mathematics work.
However, we see the inherent challenges and frustrations of problem solving as an integral part of learning mathematics. It is through cognitive dissonance
that we build strategies for problem solving and
deepen our understanding of mathematics.
Several teachers commented that, as a result of
participating in the workshop, they would now be
more understanding of their students’ struggles with
mathematics. Even with persistent questioning, however, we had difficulties eliciting responses about the
mathematics content. When we asked the teachers
to focus on what finally made the “light bulb turn
on,” most referred to the use of the manipulatives.
This led to a discussion of their initial reluctance to
actually manipulate the cubes rather than use them
merely to display a value, a reluctance similar to
some students’ reluctance to use manipulatives to
help explore mathematical situations. These stateTeaching Children Mathematics / April 2007
ments reinforce the difference between having and
using manipulatives in the mathematics classroom.
As teachers, we should not assume that because we
provide a set of linking cubes for students to use during an activity they will actually use them. Students
might be hesitant to use the cubes, as the teachers in
our workshop were, thinking that the need to work
with cubes is a weakness.
Conclusions
We found that using the Orpda number system
opened the eyes of these teachers and future teachers to some of the struggles their students face when
learning place-value concepts as defined by Van
de Walle (2007). Their additional insights into the
availability of manipulatives versus their actual use
during mathematics and the advantages of using
manipulatives themselves will result in better attention to their students and practice in the classroom.
In presenting the base-five Orpda number system to preservice and in-service teachers, we do not
expect them to master this system. We want them to
examine and evaluate the activities and models that
helped them understand the concept of place value
associated with the Orpda system. Most important,
we want them to look more deeply at the mathematics of place value.
We hope that teachers will remember the following critical ideas for developing an understanding
of place value:
• The concept of zero as a value and a place holder
is the underpinning of the base-ten system.
• The use of manipulatives, grouping activities,
and multiple representations allows exploration
of place-value concepts
• Patterns in the hundreds chart reveal characteristics of the base-ten system.
• A number’s position determines its value.
Teachers with a deeper understanding of place
value, we feel, will prepare activities for their own
students that will better help them explore place
value. They will have a more empathetic understanding of the students’ difficulties and of how to assist
them in overcoming these difficulties.
Reference
Van de Walle, John A. Elementary and Middle School
Mathematics: Teaching Developmentally. 6th ed.
Boston: Pearson Education, 2007. s
437