Handbook of psychology volume 7 educational psychology

MODELING PERSPECTIVES

In this section we conclude with an abbreviated tour of some

of the emerging work in mathematical modeling in K–12

education. Model-based reasoning is erected on foundations

of analogy, representation, and inscription. Analogies, of

course, are at the heart of modeling (Hesse, 1965). One sys-

tem stands in for another. Models are sustained by mappings

between the representing and represented worlds, and the

nature of these mappings is governed by systems of inscrip-

tion and notation (Hestenes, 1992). Consider that maps high-

light and preserve some aspects of the world while sacrificing

others. The familiar Mercator projection facilitates naviga-

tion but distorts area of landmasses, often with devastating

political consequences.

Bridging Epistemologies

The separation of world and model constitutes a bridge be-

tween the epistemologies of mathematics and science. On the

one hand, modeling provides opportunities to create coherent

and valid mathematical structures. These invite proof so that

one can better understand the edifice one is erecting for

Modeling Perspectives

383

purposes of representation (Hodgson & Riley, 2001). On the

other hand, because models and their referents are distinct,

their relation is not one of copy, but rather of fit. Fits between

models and worlds are never congruent (the separation be-

tween model and world just mentioned), so residuals between

them can be determined only in light of other potential mod-

els (Grosslight, Unger, & Smith, 1991; Lesh & Doerr, 1998).

These qualities of models and modeling practices are at the

heart of professional practice in science and in some branches

of mathematics (Giere, 1992; Stewart & Golubitsky, 1992). 

Research studies of student modeling have generally fol-

lowed two somewhat overlapping paradigms. The first line of

inquiry focuses on model-eliciting problems (Lesh, Hoover,

Hole, Kelly, & Post, 2000), in which students invent, revise,

and share models as solutions to single problems that typi-

cally are solved during one lesson. Problems are often drawn

from realms of professional practice, especially engineering,

business, and the social sciences. Consequently, they often

require students to integrate multiple forms of mathematics,

not simply the application of a single solution procedure (see

the volume by Doerr & Lesh, in press). The second line of

inquiry complements the first by engaging students in the

progressive mathematization of nature. For example,

students pose questions about motion or biological growth

and develop models as explanations (diSessa, 2000; Kaput,

Roschelle, & Stroup, 2000; Lehrer & Schauble, 2002). This

second strand of research focuses on long-term development

of student reasoning because acquiring capabilities and

propensities to adopt a modeling stance toward the world is

an epistemology with slow evolution. Consequently, research

typically spans months or even years of student learning.

Both strands of work emphasize the development of model-

based reasoning in contexts designed to support these prac-

tices, so they typically require substantial programs of

teacher professional development (e.g., Clark & Lesh, 2002;

Lehrer & Schauble, 2000; Schorr & Clark, in press). We

focus on model-eliciting problems in the remainder of this

section to represent this broader spectrum of research.

Cycles of Modeling

At the heart of a modeling perspective is the belief that some

of the most important “big ideas” in elementary mathematics

are models (or conceptual systems) for making sense of

mathematically significant types of situations (Doerr & Lesh,

2002). Model-eliciting problems are designed to evoke math-

ematical systems, not single procedures, as solutions (Lesh

et al., 2000). Students make sense of these situations not all

at once, but rather in a cycle of invention and revision

(Lesh & Doerr, 1998; Doerr, Post, & Zawojewski, 2002;

Lesh & Harel, in press). Figure 15.3 displays one such

Statement of the Big Footprint Problem

Early this morning, the police discovered that, sometime late last night, some

nice people rebuilt the old brick drinking fountain in the park.  The mayor

would like to thank the people who did it.  But, nobody saw who it was.  All

the police could find were lots of footprints. … You’ve been given a box

(shown below) showing one of the footprints.  The person who made this

footprint seems to be very big.  But, to find this person and his or her friends,

it would help if we could figure out how big the person really is?

Your job is to make a “HOW TO” TOOL KIT that

the police can use to figure out how big people are -

just by looking at their footprints.  Your tool kit

should work for footprints like the one that is shown

here.  But it also should work for other footprints.

A Brief Summary of a Big Foot Transcript

Interpretation #1– based on qualitative reasoning:  For the first 8 minutes of

the session, the students used only global qualitative judgements about the

size of footprints for people of different size and sex – or for people wearing

different types of shoes. e.g.,  … Wow! This guy’s huge. …  You know any

girls that big?!! …  Those’re Nike’s. - The tread’s just like mine.

Interpretation #2 – based on additive reasoning: One student put his foot

next to the footprint.  Then, he used two fingers to mark the distance

between the toe of his shoe and the toe of the footprint.  Finally, he moved

his hand to imagine moving the distance between his fingers to the top of his

head.  This allowed him to estimate that the height of the person who made

the footprint. But, instead of thinking in terms of multiplicative proportions

(A/B=C/D), using this approach, the students were using additive

differences.  That  is, if one footprint is 6” longer than another one, then the

heights also were guessed to be 6” different.

Note:   At this point in the session, the students’ thinking was quite unstable.

For example, nobody noticed that one student’s estimate was quite

different than another’s; and, predictions that didn’t make sense were 

simply ignored. … Gradually, as predictions become more precise,

differences among predictions began to be noticed; and, attention

began to focus on answers that didn’t make sense. Nonetheless,

“errors” generally were assumed to result from not doing procedure

carefully – rather than from not thinking in productive ways.

Interpretation #3 – based on primitive multiplicative reasoning: Here,

reasoning was based on the notion of  being “twice as big”.  That is, if my

shoe is twice as big as yours, then I’d be predicted to be twice as tall as you.”

Interpretation #4 – based on pattern recognition: Here, the students  used a

kind of concrete graphing approach to focus on trends  across a sequence of

measurements. That 

is, 

they 

lined 

up

against a wall and used footprint-to-

footprint comparisons to make estimates

about height-to-height relationships as

illustrated in the diagram shown here. …

This way of thinking was based on the

implicit assumption that the trends should

be LINEAR - which meant that the

relevant relationships were unconsciously

treated as being multiplicative.  The

students said:

Here, try this… Line up at the wall… Put your heels here

against the wall…. Ben, stand here.  Frank, stand here….  I’ll

stand here ‘cause I’m about the same (size) as Ben. {She

points to a point between Ben and Frank that’s somewhat

closer to Ben}… {pause} …  Now, where should this guy be? -

Hmmm. {She sweeps her arm to trace a line passing just in

front of their toes.}… {pause} … Over there, I think. - {long

pause}… Ok.  So, where’s this guy stand? …  About here.

{She points to a position where the toes of everyone’s shoes

would line up in a straight line.}

Note: At 

this point in the session, all three students were working

together to measure heights, and the measurements were

getting to be much more precise and accurate than earlier in

the session.

Interpretation #5: By the end of the session, the students were being VERY

explicit about comparing footprints-to-height. That is, they estimated that:

Height is about six times the size of the footprint.  For example, they say:

Everybody’s a six footer!   (referring to six of their own feet.)

Figure 15.3

A modeling cycle is characterized by waves of invention and

revision.

384

Mathematical Learning

model-eliciting problem and a case of one group of seventh-

grade students’ progressive efforts to make sense of this situ-

ation. This case illuminates several features of modeling that

have been replicated across widely diverse populations of

students and problem types (see Doerr & Lesh, 2002, for a

compendium of studies).

First, student modeling generally occurs through a series

of develop-test-revise cycles. Each cycle involves somewhat

different ways of thinking about the nature of givens, goals,

and possible solution steps (Lesh & Harel, in press). Refine-

ment typically occurs as students attempt to create coherent

and consistent mappings between the representing and repre-

sented worlds, often by noticing the implications of a particu-

lar choice of representation for the world or by noticing how a

feature of the world remains unaccounted for in a model sys-

tem. These observations resulted in several different interpre-

tations across time. Second, recalling our earlier descriptions

of inscriptional mediation of mathematical thinking, model-

ing nearly always involves multiple and interacting systems

of inscription and notation as students grapple with potential

correspondences between the world and the emerging mathe-

matical description. Third, there nearly always seem to be

multiple and often uncoordinated ideas “in the air” during

early phases of modeling, and these are reconciled and stabi-

lized as students attempt to fit models to what they consider

data. Thus, data and models often codevelop. Fourth, initial

efforts to establish fit are nearly always local, and it is the

need to consider others’ models and data that often prompts

testing for more general structures (Lesh & Doerr, 2002). In

the case outlined in Figure 15.3, the group’s model was tested

further in light of data gathered and models proposed by other

student-modelers. This underscores an expanded sense of

mathematical argument as conviction and experiment.

Although this chapter is aimed primarily at student learn-

ing, modeling research has prompted the development of new

forms of research practices and research design methodolo-

gies (Kelly & Lesh, 2000). These arose to enable multiple re-

searchers at distant research sites to coordinate work that

employed distinct theoretical perspectives focused on multi-

ple levels of interacting participants (students, teachers,

curriculum designers, and researchers; Lesh, in press). Cross-

site collaborations were accomplished by using shared tasks

and research tools (Lesh, Hoover, Hole, Kelly, & Post, 2001)

and by recognizing that all of the relevant participants (re-

searchers, teachers, students, and others) can all be thought of

as being in the business of designing models and accompany-

ing conceptual tools to make sense of their experiences. Thus,

multiple tiers of analysis were required for the conduct of

these studies (Lesh, 2002), a prospect that augments the de-

sign study perspective described previously.

IMPLICATIONS

Mathematical thinking is a specialized form of argument and

inscription, but it has its genesis in the development of every-

day capacities of pretense, possibility, conversation, and

inscription. Development of mathematical literacy relies on the

design of learning niches that support its continued evolution.

Schooling provides an unparalleled opportunity to nurture

mathematical thinking because it is one of the few arenas where

histories of learning can be systematically supported. Of

course, this opportunity is founded on the material support of

curriculum, the commitment of teachers as professionals, and

the development of knowledge about student thinking and

learning in contexts where argument and inscription take cen-

ter stage. With this in mind, we suggest a few plausible direc-

tions for research in mathematics education.

First, we urge consideration of a broader scope of mathe-

matics as worthy of research. Most studies focus on analysis

(in later grades) and number concepts (in earlier grades).

Although we believe this research has proven productive and

valuable, it ignores realms of mathematics that may well

prove foundational for a mathematics education. For exam-

ple, the Elkonin-Davydov approach to elementary mathemat-

ics education in Russia takes measurement, not “natural”

numbers, as foundational. Hence, in this program children’s

early mathematical experiences are oriented toward measure,

not count. Other possibilities suggest themselves, such as

early and prolonged emphasis on space and geometry and

consideration of the roles of modeling and design in the for-

mation of mathematical expression and epistemology.

Second, and following from a broader scope of inquiry, the

nature and grounds of professional practice in the community

of researchers require fundamental change. Study of the de-

velopment of mathematical thinking, rather than piecemeal

attention to relatively small components, requires considerate

crafting of mathematical experience so that learners consis-

tently participate in mathematical argument and expression. In

addition, it requires research designs that are coordinated with

this craftsmanship to come to understand long-term develop-

ment. The complexity of this problem suggests a reorganiza-

tion of professional practices so that design of learning

environments and study of development can be systematically

examined and become coconstituted. This form of research is

practiced currently in engineering professions, with their em-

phasis on design prototypes and iterative design. However, to

our knowledge, only embryonic forms of this way of working

currently exist in mathematics education research.

Third, and in concert with the previous two suggestions,

the focus on a mathematics education needs to be coordinated

with other realms of endeavor, recalling that the same child

References

385

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also learning to participate in scientific argument, historical

argument, and so on. Each of these forms of literacy has

implications for the development of identities and interests,

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(1938) suggested that identities and interests emerge from

personal experience and expression. Hence, if we aim to pro-

mote students as authors of mathematical expression, then we

need to understand more about how these experiences (which

are the objects of instructional design) are coordinated and

differentiated during the course of education.

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