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Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding

Beneath the
Tip of the
David C. Webb, Nina Boswinkel,
and Truus Dekker
David C. Webb,, is
an assistant professor of mathematics edu-
cation at the University of Colorado at Boul-
der. He is interested in the design of profes-
sional development that supports teachers’
classroom assessment practices. Nina Boswinkel,, and Truus Dekker,, are colleagues at the Freudenthal Institute for Science and Mathematics
Education at Utrecht University in the Netherlands. They are interested in curriculum design
and professional development focusing on more accessible instructional interventions.
Using Representations
to Support Student
Copyright © 2008 The National Council of Teachers of Mathematics, Inc. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
A common challenge for middle-
grades mathematics teachers is to
find ways to promote student under-
standing of mathematics. When an
algorithm for adding or subtracting
fractions is explained as clearly as pos-
sible and students have opportunities
to practice it, the reality is that many
students will continue to confuse the
procedures and forget how they work.
The primary intervention used to
address student confusion is to reteach
common procedures and give ad-
ditional practice in the hope that the
students will understand over time.
Over the past decade, the reform
of mathematics teaching and learning
and the implementation of NSF-
funded, Standards-based instruc-
tional materials have broadened the
options for teaching and learning
mathematics. Ideally, these new in-
structional materials should increase
opportunities for all students to learn
mathematics. However, both regular
and special education teachers have
a difficult time making instructional
decisions to address the needs of
underachieving students. In particu-
lar, to address the instructional needs
of students who are already multiple
grade levels behind, teachers need to
make informed decisions regarding
the selection of experiences that can
close the achievement gap. However,
without a framework for selecting
essential representations and experi-
ences, teachers are dependent on the
design and sequencing of activities in
the instructional materials they use in
their classrooms.
How can teachers use representa-
tions to increase student access to
mathematics? In this article, we will
describe how an “iceberg model,”
developed by the Freudenthal Insti-
tute for teachers in the Netherlands,
was used by middle-grades teachers
in the United States. This model
supports the selection of more ac-
cessible instructional interventions
and student-centered instructional
sequences. (For more information
about the Dutch approach to math-
ematics education, see Case 2006.)
Researchers at the Freudenthal In-
stitute for Science and Mathematics
Education at the University of Utrecht
developed the iceberg model to sup-
port teacher thinking about learn-
ing processes and strategies used by
students (Boswinkel and Moerlands
2001). This model has proved to be a
powerful metaphor for illustrating how
students need to experience a broad
range of mathematical models to make
sense of formal mathematical repre-
sentations (i.e., the tip of the iceberg).
A rich professional development
experience can be based on the con-
struction of representational icebergs.
Groups of teachers can work together
with the iceberg model, which offers
a context for exploring and discuss-
ing essential representations and
the sequencing of activities in their
instructional materials. This model
is a metaphor, distinguishing the role
of informal, preformal, and formal
representations used by students.
See figure 1 for an illustration of
the iceberg model. It consists of the
tip of the iceberg and a much larger
area underneath, called the float-
ing capacity. The tip of the iceberg
represents the targeted formal proce-
dure or symbolic representation. The
bulk of the iceberg that sits under the
water is represented by a combination
of informal, including context-bound,
representations (e.g., time, coins, part
of an apple, and so on), which transi-
tion to preformal strategies and mod-
els (e.g., fraction strips, a number line,
and so on). This metaphor can be used
across many different problems. In
the case of operations with fractions,
for example, even before students can
meaningfully discuss various strategies,
they need to have an understanding of
what a fraction represents.
In general, the progressive for-
malization suggested by the different
levels within the iceberg’s floating
capacity implies that more formal
representations build on less formal
ones. However, this does not mean
that as soon as a student has reached
a formal understanding that he or she
will never return to the use of prefor-
mal representations. Rather, a student
should be able to revisit preformal
Fig. 1 The iceberg model
representations, especially when new
and unfamiliar contexts are encoun-
tered. In fact, it is reasonable to expect
that some special education students
may not understand the formal rep-
resentation at all, and yet they may be
able to solve problems using informal
or preformal approaches.
Mathematics teachers are quite famil-
iar with formal representations and
strategies for manipulating them. The
common algorithms, which demon-
strate efficiency and fluency, are the
formal strategies for numeric com-
putation. Unfortunately, such formal
strategies are often presented in ways
that require students to make connec-
tions to other strategies or representa-
tions, which is difficult for students
who struggle with different models.
One example of a formal repre-
sentation would be a proportion with
one unknown (e.g., x/6 = 10/18). This
type of proportion suggests a formal
and somewhat efficient solution strat-
egy: the cross-multiply-and-divide
method. Imagine students who have
not had experience using preformal
representations, such as a ratio table,
percent bar, or double number line. In
the cross-multiply-and-divide meth-
od, there are few opportunities for
these students to relate the proportion
to other meaningful strategies that
promote number sense and propor-
tional reasoning (e.g., 6 is one-third of
18, so x should be one-third of 10).
For students, diagrams and expla-
nations are informal representations,
which are often grounded in student
experiences with real or imagined
contexts. An informal explanation for
adding “three-quarters” and “one-half
would be to reason, “Two quarters and
a half dollar is one dollar. This leaves
one quarter. So it’s a dollar and a quar-
ter.” An informal explanation using pat-
terns, for example, might relate odd and
even numbers to a V-pattern. When
students are asked, “Will a V-pattern
ever have an even number?” a student
might reason informally using the V-
formation used by geese. “On each side
of the V, the geese will have a partner,
but the leader will always be on its own.
So the number will always be odd.”
Preformal representations build on
students’ informal representations, or
reasoning, and offer greater mathe-
matical structure. Some examples are a
recursive formula to describe numeri-
cal patterns; a double number line to
solve a problem involving scale; and
an area model for multiplying whole
numbers, mixed fractions, or binomi-
als. Most preformal representations
are rarely developed by students on
their own to solve a problem. Instead,
students are guided by teachers or in-
structional materials to use preformal
representations and strategies that can
be applied across many situations and
contexts. Preformal representations
offer greater opportunities to empower
students’ sense-making, but they often
have limitations in the scope of prob-
lems that can be solved using the cho-
sen representation. For example, the
area model helps students understand
how multiplication of mixed numbers
relates to the distributive property, but
it is not very practical for computing
Some students arrive at the formal
level faster than others. However,
students should not be forced to use
formal strategies if they have not had
experiences with essential informal and
preformal representations. The time
invested in sense-making experiences
at the preformal level will substantially
reduce the time needed to reteach and
practice at the formal level.
Although professional development
experiences in which teachers iden-
tify informal, preformal, and formal
strategies and notations have been
used extensively in the Netherlands,
the mathematical iceberg activity was
recently introduced to middle-grades
mathematics teachers in the United
States. The purpose of this activity
is to encourage teachers to reflect on
the representations found in their
curricula and influence the selection
of potential instructional interven-
tions. The activity could be used for
almost any mathematical topic that
involves multiple representations,
models, or strategies.
First, teachers begin with a diagram
of an empty iceberg containing a water
line. Then, the formal representation
being studied is written in the tip of the
iceberg, above the water line (for
example, 3/4 to represent the use of
fraction notation; x/12 = 9/20 to
represent solving proportions; or y =
mx + b to represent the understanding
of linear functions). Teachers are then
asked to recall representations that their
students have used and develop a
collection of related informal and
preformal representations that contrib-
ute to student understanding of the
formal representation. This is followed
by a scavenger hunt through district-
adopted textbooks and other instruc-
tional materials to identify other
informal and preformal representations.
The heart of the iceberg activity
involves teachers working together
to identify related representations
and strategies and discussing how
these representations support student
understanding. Also, teachers discuss
building on less formal understanding
and deciding whether the representa-
tions are best categorized as informal,
preformal, or formal. Given the scope
of content that some topics involve,
teachers will also need to consider
whether a representation is distinctive
enough to merit its own place in the
iceberg. Depending on the mathemati-
cal representation at the tip of the ice-
berg, this activity might take anywhere
from 45 to 120 minutes to complete.
Building on an iceberg model cre-
ates a context for a lively debate about
representations, their connections to
students’ prior knowledge, and how
collectively the teachers support the
students’ understanding of mathemat-
ics. It is also important to recognize
questions about the usefulness of par-
ticular representations or strategies, and
their potential drawbacks, that might
emerge during group discussions. Time
should be set aside to deliberate these
real (or perceived) limitations.
The goal of constructing math-
ematical icebergs is to summarize the
teachers’ collective knowledge of rep-
resentations and how those representa-
tions are interrelated. Formal represen-
tations are often considered the only
important (end) goal. Unfortunately,
they receive inordinate attention in
classroom assessments even though less
formal representations are invaluable in
assessing a student’s prior knowledge
and revealing potential starting points
for instruction and intervention.
An indispensable follow-up activity to
constructing icebergs involves review-
ing instructional materials for repre-
sentational pathways. For example, as
part of a summer workshop focused
on the development of rational
number concepts and proportional
reasoning, middle-grades teachers
were given cards that included activi-
ties and representations from the two
most commonly used mathematics
curricula in the district. “Formal Ad-
dition of Fractions” was the concept
named at the tip of the iceberg.
Groups of teachers first discussed how
to organize the cards vertically within
an iceberg to represent the progressive
formalization of the adding fractions.
Then the groups deliberated over the
appropriate sequencing of activities,
offering suggestions for extra cards
when they thought that something
important was missing or deleting
superfluous cards (see fig. 2).
Teachers found that the represen-
tational pathways in their instruc-
tional materials were either exclusively
formal representations (found in one
textbook series) or a mix of represen-
tations below the water line (as seen
in another textbook series). Teachers
also found that some groups con-
structed representational pathways
that were dramatically different from
their own, which led to a substantive
discussion of the appropriate ordering
of activities and options for alternative
pathways. Another observation was
that important representations seemed
to be missing from the available
cards, and so teachers were invited to
describe what these missing activities
should be. Crucial learning experienc-
es in the pathway, which should not
be skipped, were marked by teachers
with a red box. A next step would be
to add activities to the representation-
al pathways around crucial learning
experiences that seemed to be missing
in the instructional materials.
When developing instructional
plans for students who need individu-
alized interventions, the representa-
tional pathways help teachers identify
the appropriate starting points based
on a student’s prior knowledge.
Although the Dutch project focused
on supporting teachers of special
education students, the construction
and application of iceberg models and
representational pathways is useful to
teachers of all students.
With respect to opportunities for
teacher learning, this article is only
the “tip of the iceberg” for the types
of professional development activi-
ties that can be done using multiple
representations. Such development
can support collaborative instructional
planning, curriculum mapping, and
identification of appropriate interven-
tions for struggling students.
Boswinkel, N., and F. Moerlands. Het topje
van de ijsberg [The top of the iceberg].
In De Nationale Rekendagen, een
praktische terugblik [National confe-
rence on arithmetic, a practical view].
Utrecht: Freudenthal Institute, 2003.
Case, Robert W. “Report from the
Netherlands: The Dutch Revolution
in Secondary School Mathematics.”
Mathematics Teacher 98 (February
2005): 374–82.
Fig. 2 Teachers working on the design
of representational pathways
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  • Robert W Case
Utrecht: Freudenthal Institute, 2003. Case, Robert W. "Report from the Netherlands: The Dutch Revolution in Secondary School Mathematics." Mathematics Teacher 98 (February 2005): 374-82