deeping teachers understanding of place value.pdf

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434 Teaching Children Mathematics /April2007

Theresa M. Hopkins, [email protected], is a postdoctoral fel-

low 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 in-

service teachers and also in rural mathematics education. Jo

Ann Cady, [email protected], is an associate professor of math-

ematics 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.

By Theresa M. Hopkins and Jo Ann Cady

As aculty members o the Mathematics Edu-

cation Group in the College o Education,

Health, and Human Sciences at the Univer-

sity o Tennessee, we are responsible or instructing

both preservice and in-service teachers through

courses and proessional development activities.

One topic we address is teaching place value to

elementary school students. Teachers amiliarity

with the base-ten number system, however, can

prevent them rom ully comprehending the di-

Deepening Teachers Understanding of Place Value

@*#?WHAT IS THE VALUE OF

Copyright 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reservedThis material may not be copied or distributed electronically or in any other format without written permission from NCTM


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Teaching Children Mathematics /April2007 435

culty these students have when trying to understand

the abstract concept o place value. This article

presents our evolving lesson in addressing this

diculty.

The Signifcance oUnderstanding Place ValueTo circumvent our mathematics education students

amiliarity with the base-ten number system, we had,

in the past, used base-ve blocks to investigate place

value. However, rather than working within the base-

ve system, many o our students simply tried to

convert rom base-ve to base-ten. Further, both our

preservice and in-service teachers continually read

numbers incorrectlyor example, they read 105

as

10 rather than as one zero base-ve. This real-ization led us to brainstorm ideas or activities that

would produce a cognitive dissonance in these teach-

ers, orcing them to think about place-value concepts

such as these identied by Van de Walle (2007):

1. Sets o ten (and tens o tens) can be perceived

as single entities. These sets can then be

counted and used as a means o describing

quantities. For example, three sets o ten and

two singles is a base-ten method o describ-

ing 32 single objects. This is the major prin-

ciple obase-ten numeration.

2. The positions o digits in numbers determine

what they representwhich size group they

count. This is the major principle oplace-

value numeration.

3. There are patterns to the way that numbers

are ormed.

4. The groupings o ones, tens, and hundreds

can be taken apart in dierent 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

fexible ways is a signicant skill or com-

putation. (p. 187)

The Orpda Number SystemBecause o the diculties students conront in

understanding place value, we as acilitators

decided that we would create a new number sys-

temwhich we named Orpdathat would use

symbols rather than numerals to represent values.

This approach had the advantage o making the

activity more abstract so that the teachers experi-

ence with place value would be similar to that o

their elementary school students introduction to

place value. In previous workshops and courses

that we had conducted, the teachers amiliarity

with base-ten numerals oten interered with their

learning, but this approach intentionally placedthem into an entirely new number system. We also

wanted the teachers to explore the importance o

concrete models and grouping activities, so we

provided multilink cubes or early sessions and

base-ve blocks or later sessions.

To begin the activity, we placed a blank transpar-

ency 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

o objects inside the circle. The ensuing discussion

included our statement that ~ represented nothing.

Then sets containing 1, 2, 3, and 4 objects wereillustrated and represented by dierent symbols as

shown in table 1. To be sure that the teachers were

comortable working with the Orpda symbols, we

asked them to indicate the symbol that represented

several dierent sets o objects ranging in number

rom 0 to 4 objects.

We explained that these ve 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 teach-

ers might need some time to discuss answers to this

question within their groups, we were surprised tosee several hands go up immediately, but we allowed

a ew minutes or others to think about the question

beore soliciting responses. Although we expected

the answer *~, representing a single group o 5 and

no units, several teachers gave us alternative answers

that quickly pushed all o us, teachers and acilitators

alike, out o our comort zone.

The rst 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; thereore, *& represented 5. We

Number o Objects and Representative

Symbol in the Orpda Number System

No objects

~1 object *2 objects @

3 objects #

4 objects &

Table 1


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436 Teaching Children Mathematics /April2007

acknowledged her answer, recorded it on the board,

and then asked i anyone had a dierent one. A sec-

ond teacher gave the answer @#. Again, we wrote

the answer on the board and asked or an explana-

tion. The teacher replied that @ represented 2 and #represented 3, or a sum o 5. These answers were

unexpected, but even more unexpected was that no

one in the group o nearly ty teachers suggested

what we considered the correct answer.

Rather than immediately commenting on these

answers, we began a discussion about how to repre-

sent the value 6. This gave us time to think about a

new direction and urther evaluate the participants

mathematical thinking. Again, the teachers gave

several dierent answers, some using two symbols

and some using three, such as @@@. They dis-

cussed the validity o the various representationsand concluded that all were valid. We accepted these

justications and acknowledged their reasonableness

but then posed this question: I we have many ways

o representing the same number, how would we

know which to use? We countered the explanation

that @# represented the value 5 with an example

rom 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 o represent-

ing values greater than 4, using the Orpda number

system would be very conusing.

Relating to the base-ten systemWe asked the teachers to think about the numerical,

or place-value, relationships in the base-ten number

system and then create a unique symbolic repre-

sentation o 5 in the Orpda number system. Ater

much discussion within their groups, the teachers

suggested the answer *~, which we displayed on the

board. Several teachers were skeptical and asked or

an explanation. The teacher who had suggested this

answer compared *~ to 10 in the base-ten system,

saying we had one group o 5 with no singles let

over; this ormulation compares with one group o 10

with no singles in the base-ten system (see fg. 1).

Many o the teachers accepted this representationand its explanation, but some did not. A discussion

regarding the use o ~ (zero) ensued. Several teach-

ers suggested that ~ represented nothing, as dened

at the beginning o the lesson, and should not be

used. Others countered that ~ represented the idea

that there were no units but rather one group o 5

objects, hence the use o *~ to represent the value

displayed. They maintained that their idea was

related to the idea o 10 in the base-ten system being

represented by one group o 10 objects and no units

or ones. We then provided more inormation on the

invention o zero and its use as a place holder. Theteachers struggles with the concepts o zero, group-

ing, and the position o digits within a numeral par-

allel the struggles their elementary school students

might have with the same concepts.

Using manipulatives tounderstand a number systemUsing multilink cubes, the teachers were then asked

to count and represent sets o objects up to 30 within

the Orpda number system. We ound it interesting

that the teachers used the multilink cubes to repre-

sent the number o objects but did not use them to

create groups o 5. As we continued to count, the

teachers began to see the patterns that developed,

similar to patterns that elementary school students

nd when completing a hundreds table.

Several teachers connected the ideas rom the

base-ten number system to the Orpda number system

and quickly created symbols to represent the values

o sets o objects through *~ ~ (25). Others needed

more time to think and some guidance rom the acil-

itators. We let the teachers struggle with the problem

Combining units to create a rod in the base-ten number system (a) and the Orpda (base-fve)

number system (b)

Figure 1

+ +

1a. 10 = 1 rod and 0 units 1b. *~ = * rods and ~ units


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Teaching Children Mathematics /April2007 437

on their own until we elt the rustration level rise;

only then did we step in to oer suggestions to those

who were rustrated to the point o no longer trying.

At this point, we suggested that they use the cubes to

model the values. Rather than show or tell the teach-ers what to do with the multilink cubes, we contin-

ued to ask questions such as these:

What do you already know about the symbols

or numbers in the Orpda system?

How can you represent these symbols with

cubes?

How would a set o X X X X X X objects be

represented in Orpda?

What would happen i 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?

Ater a short time, one teacher suggested that we

create a group o ve rods, or groups o 5, to make

a square (fat) and use the symbol *~ ~ to represent

this value. Many o 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 o 25 (5 fats).

At the conclusion o many o our workshops,

we encouraged a discussion about the participants

rustration level. We were surprised by the teachers

vehement opposition to using the wordfrustration to

describe eelings 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 rustra-

tions o problem solving as an integral part o learn-

ing mathematics. It is through cognitive dissonance

that we build strategies or problem solving and

deepen our understanding o mathematics.

Several teachers commented that, as a result o

participating in the workshop, they would now bemore understanding o their students struggles with

mathematics. Even with persistent questioning, how-

ever, we had diculties eliciting responses about the

mathematics content. When we asked the teachers

to ocus on what nally made the light bulb turn

on, most reerred to the use o the manipulatives.

This led to a discussion o 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 state-

ments reinorce the dierence between having and

using manipulatives in the mathematics classroom.

As teachers, we should not assume that because we

provide a set o linking cubes or students to use dur-

ing an activity they will actually use them. Studentsmight be hesitant to use the cubes, as the teachers in

our workshop were, thinking that the need to work

with cubes is a weakness.

ConclusionsWe ound that using the Orpda number system

opened the eyes o these teachers and uture teach-

ers to some o the struggles their students ace when

learning place-value concepts as dened by Van

de Walle (2007). Their additional insights into the

availability o manipulatives versus their actual useduring mathematics and the advantages o using

manipulatives themselves will result in better atten-

tion to their students and practice in the classroom.

In presenting the base-ve Orpda number sys-

tem 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 o place value

associated with the Orpda system. Most important,

we want them to look more deeply at the mathemat-

ics o place value.

We hope that teachers will remember the ollow-

ing critical ideas or developing an understanding

o place value:

The concept o zero as a value and a place holder

is the underpinning o the base-ten system.

The use o manipulatives, grouping activities,

and multiple representations allows exploration

o place-value concepts

Patterns in the hundreds chart reveal characteris-

tics o the base-ten system.

A numbers position determines its value.

Teachers with a deeper understanding o placevalue, we eel, will prepare activities or their own

students that will better help them explore place

value. They will have a more empathetic understand-

ing o the students diculties and o how to assist

them in overcoming these diculties.

ReerenceVan de Walle, John A. Elementary and Middle School

Mathematics: Teaching Developmentally. 6th ed.Boston: Pearson Education, 2007. s

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