Concrete Materials and Teaching for Mathematical Understanding
Author(s): Patrick W. Thompson and Diana Lambdin
Source: The Arithmetic Teacher , MAY 1994, Vol. 41, No. 9 (MAY 1994), pp. 556-558
Published by: National Council of Teachers of Mathematics
Stable URL: https://www.jstor.org/stable/41196106
REFERENCES
Linked references are available on JSTOR for this article:
https://www.jstor.org/stable/41196106?seq=1&cid=pdfreference#references_tab_contents
You may need to log in to JSTOR to access the linked references.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide
range of content in a trusted digital archive. We use information technology and tools to increase productivity and
facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
https://about.jstor.org/terms
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve
and extend access to The Arithmetic Teacher
This content downloaded from
110.144.70.46 on Mon, 15 Feb 2021 10:40:56 UTC
All use subject to https://about.jstor.org/terms
RESEARCH
INTO
PRACTICE
Concrete Materials and Teaching for Mathematical
Understanding
students to understand?" Too often it is
Learning without thought is labor lost.
- Confucius
ers would not necessarily benefit from them.
"What shall I have my students learn toHowever, Suydam and Higgins (1977) re-
do?" If we can answer only the second
ported a pattern of beneficial results for all
question, then we have not given sufficient
learners. Middle and upper primary stu-
An experience is not a true experience until
it is reflective.
thought to what we hope to achieve by dents
a
observed by Labinowicz (1985) ex-
- John Dewey
particular segment of instruction or use perienced
of
considerable difficulties making
concrete materials.
sense of base-ten blocks, although Fuson
we find common agreement that
and Briars ( 1 990) reported astounding suc-
effective mathematics instruction in
cess in the use of the same materials in
the elementary grades incorporates liberal
teaching addition and subtraction algo-
use of concrete materials. Articles in the
rithms. Thompson (1992) and Resnick and
Omanson (1987) reported that using base-
Arithmetic Teacher no longer exhort us
to use concrete materials, nor does the
Professional Standardsfor Teaching Math-
ematics (NCTM 1991) include a standard
How shall we
ten blocks had little effect on upper-
think about this
on the use of concrete materials. The use of
primary students' understanding or use of
their already memorized whole-number
addition and subtraction algorithms,
whereas Wearne and Hiebert (1988) reported consistent success in the use of
problem?
concrete materials seems to be assumed
unquestioningly.
concrete materials to aid students' under-
My aim in this article is to reflect on the
role of concrete materials in teaching for
mathematical understanding - not to arResearch
gue against their use but instead to argue
Concrete
for using them more judiciously and reflec-
standing of decimal fractions and decimal
numeration (see also Hiebert, Wearne, and
on the Use of
Materials
Taber [1991]).
These apparent contradictions are prob-
tively. Our primary question should alably due to aspects of instruction and stuThe use of concrete materials has always
ways be "What, in principle, do I want my
dents' engagement to which studies did not
been intuitively appealing. The editors of a
attend. Evidently, just using concrete ma-
turn-of-the-century methods textbook
terials is not enough to guarantee success.
Prepared by Patrick W. Thompson
stated, "Examples in the concrete are better
ban uiego ótate university
for the student at this stage of his develop-
We must look at the total instructional
environment to understand effective use of
San Diego, CA 92120
ment, as he can more readily comprehend
these" (Beecher and Faxon 1918, 47, asconcrete materials - especially teachers'
images of what they intend to teach and
quoted in McKillip et al. 1978). Their apstudents' images of the activities in which
pearance accelerated in the 1960s, at least
Edited by Diana Lambdin
Indiana University
Bloomington, IN 47405-1006
are asked to engage.
Patrick Thompson teaches at San Diego State Uni- in the United States, with the publication they
of
versity, San Diego, CA 92120. He is codirectorofthe
Quantitative Reasoning Project sponsored by the
National Science Foundation.
theoretical justifications for their use by
^^nilil^^H
Zolton Dienes ( 1 960) and by Jerome Bruner
(1961).
Preparation of this paper was supported by NationalA number of studies on the effectiveness
Science Foundation (NSF) grants no. MDR 89-503 1of
1
What does this collection
using concrete materials have been con- illustrate?
and 90-96275 and by a grant of equipment from
ducted since Dienes' s and Bruner' s publiApple Computer, Inc., Office of External Research.
cations, and the results are mixed. Fennema
Any conclusions or recommendations stated here are
(1972) argued for their use with beginning
those of the author and do not necessarily reflect
official positions of NSF or Apple Computer.
learners while maintaining that older learn556
ARITHMETIC
This content downloaded from
110.144.70.46 on Mon, 15 Feb 2021 10:40:56 UTC
All use subject to https://about.jstor.org/terms
• •too
TEACHER
Seeing Mathematical
ined later in this article.
correct interpretations of materials -
Seeing mathematical ideas in concrete namely, the one we intend they have. In
Ideas in Concrete
materials can be challenging. The material actuality the opposite is meant. It should be
Materials
may be concrete, but the idea that students our instructional goal that students can
are intended to see is not in the material.
make, in principle, all interpretations of
It is often thought, for example, that an
The idea is in the way the teacher underfigure 1.
actual wooden base-ten cube is more constands the material and understands his or
A teacher needs to be aware of multiple
her actions with it. Perhaps two examples interpretations of materials so as to hear
hints of those that students actually make.
sidering the physical characteristics of will
ob- illustrate this point.
A
common
approach
to
teaching
fracWithout this awareness it is easy to prejects, this statement certainly seems true.
tions is to have students consider a collecsume that students see what we intend they
But to students who are still constructing
see, and communication between teacher
tion
of
objects,
some
of
which
are
distinct
concepts of numeration, the "thousandness"
crete to students than is a picture of a
wooden base-ten cube. When one is con-
of a wooden base-ten cube often is no more
from the rest, as depicted in figure 1 . If aand student can break down when students
concrete than the "thousandness" of a pic- student had a collection like that depicted see something other than what we prein figure 1# it would certainly be concrete. sume.
tured cube (Labinowicz 1985). To underAlso, it is important that student
stand the cube - either actual or pictured - But what might the collection illustrate to
students?
Three
circles
out
of
five?
If
so,
create
multiple interpretations of m
as representing a numeral value of 1000,
students need to create an image of a cube they see a part and a whole but not a als. They are empowered when they
that entails its relations to its potential parts fraction. Three-fifths of one? Perhaps. But nize the multiplicity of viewpoints
(e.g., that it can be made of 10 blocks each depending on how they think of the circles which valid interpretations can be
having a value of 100, 100 blocks each and collections, they could also see three- for they are then alert to choose am
having a value of 10, or 1000 blocks each fifths of five, five-thirds of one, or five- them for the most appropriate action
tive to a current situation. However,
having a value of 1 ). If their image of a cube thirds of three (see fig. 2).
They
could
also
see
figure
1
as
illustratteacher's responsibility to cultivate
is simply as a big block named thousand,
ing that 1 h- 3/5 = 1 2/3- that within 1 view. It probably will not happen if
then no substantive difference accrues to
whole is one three-fifths and two-thirds
teacher is unaware of multiple inter
students between a picture of a cube or an
of another three-fifths, or that 5-^3 = tions or thinks that the ideas are "there" in
actual cube - the issue of concreteness
would be immaterial to their understand-
1 2/3 - that within 5 is one three and two-
the materials.
thirds of another three. Finally, they could
Figure 1 is customarily offered by texting of base-ten numeration. This comment
see
figure
1
as
illustrating
5/3
X
3/5
=1
books
and by teachers to illustrate 3/5.
is not to say that, to students, a substantive
that five-thirds of (three-fifths of 1) is 1 . It
difference never occurs between pictures
is an error to think that a particular material
and actual objects. Rather, it says only that
concrete materials do not automaticallyor illustration, by itself, presents an idea
unequivocally. Mathematics, like beauty,
carry mathematical meaning for students.
is in the eye of the beholder - and the eye
A substantive difference can exist between
how students experience actual materials
and how they experience depictions of
sees what the mind conceives.
Teachers sometimes understand the dis-
Period. In fact we rarely find textbooks or
teachers discussing the difference between
thinking of 3/5 as "three out of five" and
thinking of it as "three one-fifths." How a
student understands figure 1 in relation to
the fraction 3/5 can have tremendous con-
sequences. When students think of frac-
materials, but the difference resides in how cussion of figure 1 and figure 2 as saying tions as "so many out of so many," they are
they are used. This point will be reexam-
that we need to take care that students form
^^■^^^^^^^^^^^^^^^^^^^^^^^^^^^H
justified in being puzzled about fractions
like 6/5. How do you take six things out of
five?
The second example continues the dis-
cussion of fractions.
Various ways to think about the circles and collections
in It illustrates that how
we think of the materials in a situation can
figure 1.
have implications for how we may think
If we see • • • О О as one collection, then • is one-fifth of one,
about our actions with them.
so • • • is three-fifths of one.
Suppose, in figure 3, that the top collection is an example of 3/5 and the bottom
If we see 9 • • as one collection, then • is one-third of one,
collection is an example of 3/4. Next com-
so 0 ф # О О is five-thirds of one.
bine the two collections. Does the com-
If we see • as one circle, then • • • О Ois five circles,
3/5 + 3/4? Yes and no. If we were thinking
bined collection produce an example of
so # is one-fifth of five and • • • is three-fifths of five.
If we see • as one circle, then Ф • • is three circles,
so • is one-third of three and • • • О О is five-thirds of three.
of 3/5 and 3/4 as ratios (so many out of
every so many), then 3/5 + 3/4 = 6/9 (three
out of every five combined with three out
of every four gives six out of every nine, as
when computing batting averages). If we
understood 3/5 and 3/4 in figure 3 as frac-
MAY1994
This content downloaded from
110.144.70.46 on Mon, 15 Feb 2021 10:40:56 UTC
All use subject to https://about.jstor.org/terms
557
tions, then it doesn't make sense to talk
about combining them. It would be like
asking, "If we combine three-fifths of a
large pizza and three-fourths of a small
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^H
Three-fifths and three-fourths combined into one collection
pizza, then how much of a pizza do we
have?" How much of which kind of pizza?
It only makes sense to combine amounts
(|MO^) 3/5
<ř##5> 3/4
measured as fractions when both are measured in a common unit. Both answers
(6/9 and "the question doesn't make
sense") are correct - each in regard to a
particular way of understanding the con-
V
crete material at the outset.
Using Concrete
Materials in Teaching
Elizabeth H. "Models and Mathematics."
SuchFennema, conve
Arithmetic Teacher 1 8 (December 1972):635-40.
and
Hiebe
Fuson, Karen C, and Diane J. Briars. "Using a Basephase
of
Ten Blocks Learning/Teaching
Approach for First-
It is not easy to use concrete materials well,
and it is easy to misuse them. Several
structing
and Second-Grade Place- Value and Multidigit
Addition and Subtraction." Journal for Research
of
thinkin
studies suggest that concrete materials are
convention
180-206.
in Mathematics Education 21 (May 1990):
likely to be misused when a teacher has in
mind that students will learn to perform
notation.
Hiebert, James, Diana Wearne, and Susan Taber.
we think about this? How can we think
"Fourth Graders' Gradual Construction of Deci-
mal Fractions during Instruction Using Different
about this ? How shall we think about this?"
some prescribed activity with them
(Boyd 1992; Resnick and Omanson 1987; is also the heart of what Thompson et al.
Thompson and Thompson 1994). This situ- ( 1 994) call "conceptually oriented" instrucation happens most often when teachers tion.
use concrete materials to "model" a symSecond, concrete materials furnish somebolic procedure. For example, many teach- thing on which students can act. The pedaers use base-ten blocks to teach addition
gogical goal is that they reflect on their
and subtraction of whole numbers. Stu-
actions in relation to the ideas the teacher
has worked to establish and in relation to
dents often want to begin working with the
Physical Representations." Elementary School
Journal 91 (March 1 99 1 ):32 1 -41 .
Labinowicz, Edward. Learning from Children: New
Beginnings for Teaching Numerical Thinking.
Menlo Park, Calif.: Addison- Wesley Publishing
Co., 1985.
McKillip, William D., Thomas J. Cooney, Edward J.
Davis, and James W. Wilson. Mathematics Instruction in the Elementary Grades. Morristown,
N.J.: Silver Bürde« Co., 1978.
National Council of Teachers of Mathematics. Pro-
largest blocks, such as adding or subtractthe constraints of the tasks as they have
fessional Standards for Teaching Mathematics.
ing the thousands in two numbers. Teachconceived it. The discussion of figure 3 Reston, Va.: The Council, 1991.
ers often say, "No, start with the smallest
highlighted occasions where the teacher Resnick, Lauren, and Susan Omanson. "Learning to
blocks, the ones. You have to go from right
and the students might reflect on the
to left." Would it be incorrect to start with
the largest blocks? No, it would be uncon-
fractional amounts for what it means to
ventional but not incorrect.
combine two collections.
Since our primary question should be
"What do I want my students to under-
Understand Arithmetic." In Advances in Instruc-
implications of various understandings of
tional Psychology, vol. 3, edited by Robert Glaser,
41-95. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1987.
Suydam, Marilyn M., and John L. Higgins. ActivityBased Learning in Elementary School Mathematics: Recommendations from the Research. Columsuccessful
bus, Ohio: ERIC/SMEE, 1977.
Concrete materials can be an effective
aid to students' thinking and to
stand?" instead of "What do I want my teaching. But effectiveness is contingent
students to do?" it is problematic when
Thompson, Alba G., Randolph A. Philipp, Patrick
on what one is trying to achieve. To draw W. Thompson, and Barbara A. Boyd.
teachers have a prescribed activity in maximum benefit from students' use of
"Calculational and Conceptual Orientations
mind - a standard algorithm - and un- concrete materials, the teacher must con- in Teaching Mathematics." In Professional
Development for Teachers of Mathematics, 1994
thinkingly reject creative problem solving tinually situate her or his actions with the Yearbook
when it doesn't conform to convention or
of the National Council of Teachers of
question, "What do I want my students to Mathematics, edited by Douglas B. Aichele.
prescription. In situations in which stu- understand?"
dents' unconventional, but legitimate, actions are rejected by a teacher, students References
Reston, Va.: The Council, 1994.
Thompson, Patrick W. "Notations, Conventions, and
Constraints: Contributions to Effective Uses of
Concrete Materials in Elementary Mathematics."
Journal for Research in Mathematics Education
23 (March 1992):123-47.
learn once more that "to understand" means
Beecher, W. J., and G. B. Faxon, eds. Practical
to memorize a prescribed activity.
Methods and Devices for Teachers. Dansville,
N.Y.: F. W. Owens Publishing Co., 1918.
Thompson, Patrick W., and Alba G. Thompson.
Boyd, Barbara A. "The Relationship between Math"Talking about Rates Conceptually, Part I: A
Concrete materials are used appropriately for two purposes. First they enable
the teacher and the students to have
ematics Subject Matter Knowledge and Instruction: A Case Study." Master's thesis, San Diego
State University, 1992.
grounded conversations about something
"concrete." The nature of the talk should be
(a) how to think about the materials and (b)
the meanings of various actions with them.
558
Bruner, Jerome. The Process of Education. New
York: Vintage Books, 1961.
Dienes, Zoltan. Building Up Mathematics. London:
Hutchinson Educational, 1960.
ARITHMETIC
This content downloaded from
110.144.70.46 on Mon, 15 Feb 2021 10:40:56 UTC
All use subject to https://about.jstor.org/terms
Teacher's Struggle. Journal for Research in Math-
ematics Education 25 (May 1994):279-303.
Wearne, Diana, and James Hiebert. "A Cognitive
Approach to Meaningful Mathematics Instruction: Testing a Local Theory Using Decimal Numbers." Journal for Research in Mathematics Edu-
cation 19 (November 1988):371- 84. Щ
TEACHER