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Beneath the

Tip of the

Iceberg:

David C. Webb, Nina Boswinkel,

and Truus Dekker

David C. Webb, dcwebb@colorado.edu, 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, N.Boswinkel@ﬁ.uu.nl, and Truus Dekker,

T.Dekker@ﬁ.uu.nl, 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.

110 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 14, No. 2, September 2008

Using Representations

to Support Student

Understanding

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This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 14, No. 2, September 2008 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 111

a

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.)

AN ICEBERG FULL OF

REPRESENTATIONS

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

112 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 14, No. 2, September 2008

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.

INFORMAL AND PREFORMAL

REPRESENTATIONS

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

57

12

×31

7.

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.

CONSTRUCTING THE ICEBERG

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.

Vol. 14, No. 2, September 2008 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 113

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.

FROM ICEBERGS TO

REPRESENTATIONAL

PATHWAYS

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.

REFERENCES

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