Kaput. Representación, Original

8/3/2019 Kaput. Representacin, Original.

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Shifting Representational Infrastructures and Reconstituting Content to

Democratize Access to the Math of Change & Variation:

Impacts On Cognition, Curriculum, Learning and Teaching1

[10/16/00 Draft]

James J. Kaput

Department of Mathematics

University of Massachusetts-Dartmouth

Jeremy Roschelle

Center for Technology and Education

SRI International

ABSTRACT

In order to set the basis for examining potential impacts on teacher education,

this informal essay tracks the impact of (a) deep and historic shifts in

representational infrastructure, from formal character string-based algebraic

infrastructure towards visually definable and editable functions, (b) new dynamic

change-visualization tools and learning environments that support direct linksamong mathematical notations, simulations, and support physical data-

import/export tools. In particular, we will examine how these ingredients affect

(1) the nature of traditional mathematics of change content, (2) student thinking

and learning of both old and new content, particularly by tapping more deeply

into students' cognitive, linguistic and kinesthetic resources, (3) curriculum

structure taken as given for centuries, and (4) appropriate pedagogies.

Illustrations will be drawn from work in the authors' ongoing SimCalc Project,

which builds and tests software simulations, related visualization tools, and

1 This material is based upon work supported by the National Science Foundation under Grant No. 9619102 &

0087771. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the

author(s) and do not necessarily reflect the views of the National Science Foundation.


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curriculum and teacher-support materials intended to render more learnable and

teachable the ideas underlying calculus beginning in the early middle grades. We

will reflect on how such technologies can change the experienced nature of the

subject matter and alter assumptions regarding the appropriate structure of

curriculum that have been unchallenged for centuries. We will also reflect upon

teacher learning of newly reconstituted content, including learning in the newer

classroom contexts of networked diverse platforms running parallel software.

Introduction: The Larger Historical Perspective

Why History?

We devote space to the historical basis of our current situation because it helps us understand the

depth of existing curricular assumptions and their connections with other conditions of

mathematics and science education that, while taken as given, deserve examination. This is

especially the case in design of teacher education programs whose client-products will be teaching

students who will be working well into the 22nd century. Put differently, design of teacher

preparation needs to be more forward-oriented than almost any other educational design. But, of

course, in its current forms, it is among the most conservative aspects of our educational system,

optimized for stasis, not change. Fitness of aspiring teachers is almost always taken to be the

extent to which they can fit into the existing system of education. And their preparatory

educational experiences typically take the given curricular structures and teaching practices as thestarting point, to be improved, but not fundamentally changed. As we hope to make apparent, the

changes in representational infrastructures that provide newly visual and dynamic access to core

mathematics and that are at the heart of our work involve foundational reconstitution of the content

itself. Such a reconstitution forces re-examination of most target issues of this meeting as well as

some that are not, e.g., assessment and accountability.

The Shift From Static, Inert Media to Dynamic, Interactive Media

The systems of knowledge that form the core of what was taught in schools and universities in the

20th century were built using some representational infrastructures that evolved (e.g., alphabetic

and phonetic writing) and others that were somewhat more deliberately designed, mainly by and

for a narrow intellectual elite (e.g., operative algebra). In all cases they were instantiated in and

hence subject to the constraints of the static, inert media of the previous several millennia. But the

computational medium is neither static nor inert, but rather, is dynamic and interactive, exploiting

the great new advance of the 20th century, autonomously executable symbolic processes that is,


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operations on symbol systems not requiring a human partner (Kaput & Shaffer, in press). We see

three profound types of consequences:

Type 1 : The knowledge produced in static, inert media can become knowable and learnable in new

ways by changing the medium in which the traditional notation systems in which it is

carried are instantiated for example, creating hot links among dynamically changeablegraphs equations and tables in mathematics.

Type 2 : New representational infrastructures become possible that enable the reconstitution of

previously constructed knowledge through, for example, the new types of graphs and

immediate connections between functions and simulations and/or physical data of the type

developed and studied in the SimCalc Project to be described below.

Type 3 : The construction of new systems of knowledge employing new representational

infrastructures for example, dynamical systems modeling or multi-agent modeling of

Complex Systems with emergent behavior, each of which has multiple forms of notations

and relationships with phenomena, as discussed among several of the researchers at this

meeting.

Tracing any of these complex consequences is a challenging endeavor, particularly since they

overlap in substantive ways due to the inherent ambiguity in attempts to characterize knowledge

apart from the means by which it is represented and used. Hence we will limit our discussion to a

few cases close to our recent work in the SimCalc Project involving the Mathematics of Change &

Variation (MCV), of which a subset concerns the ideas underlying Calculus. Thus we will be

focusing on a Type 2 change.

The Case of Calculus and Its Supporting Representational Infrastructures

While the Greeks, most notably Archimedes, whose extraordinary computational ability

compensated for the weaknesses of the available representational infrastructure in supporting

quantitative computation, developed certain mainly geometric ideas and techniques, the

Mathematics of Change and Variation leading to what came to be called Calculus evolved

historically beginning with the work of the Scholastics in the 1300s through attempts to

mathematize change in the world (reviewed in Kaput, 1994). The resulting body of theory and

technique that emerged in the 17th and 18th centuries, cleaned up for logical hygiene in the 19th, is

now institutionalized as a capstone course for secondary level students in many parts of the world,

and especially in the United States. These ultimately successful attempts were undertaken by the

intellectual giants of Western civilization, who also developed the representational infrastructure of

algebra, including extensions to infinite series and coordinate graphs, as part of the task. Their

work led to profoundly powerful understandings of the different ways quantities can vary, how


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these differences in variation relate to the ways the quantities accumulate, and the fundamental

connections between varying quantities and their accumulation. These efforts also gave rise to the

eventual formalization of such basic mathematical ideas as function, series, limit, continuity, etc.

(Boyer, 1959; Edwards, 1979).

Over the past two+ centuries this communitys intellectual tools, methods and productsthe

foundations of the science and technology that we utterly depend uponwere institutionalized as

the structure and core content of school and university curricula in most industrialized countries

and taken as the epistemological essence of mathematics (Bochner, 1966; Mahoney, 1980). The

resulting historically privileged algebraic notation system for representing quantitative relationships

affords quantitatively coherent transformations, combinations and comparisons of character-

strings, usually representing closed-form descriptions of functions or relations.

Consequences for Todays Curricular Structure

The algebraic techniques developed by the masters in the 17th and 18th centuries to model rates of

change and accumulations of variable quantities have remained at the heart of the modal calculus

course to this day. The requirements of that modal calculus course govern the prerequisite

structure of much of students experience with earlier mathematicswhether or not those students

are among the 10% or fewer who will study calculus. Importantly, however, the intellectual

triumphs that yielded the web of concepts and technique at the heart of that course occurred largely

without regard to learnability outside the community of intellectual elite involved. Even Leibniz,

whose carefully crafted notations we utilize today, engaged only his peers in his notation designdecisions (Edwards, 1979). Furthermore, mastery of the algebraic prerequisites became the

measure by which academic success was defined. Mastery of these algebraic tools became the

gateway to all that academic success offers, more often than not perpetuating social class structure

that advantages some students above others in access to these prerequisites.

The fact that the basic curricular structures set down in textbooks by L'Hopital, the Bernoulli's,

Euler, and their contemporaries, have remained largely invariant through the 20th century is not

merely a matter of inertia, because these structures served traditional purposes and populations

extremely well. Indeed, this basic intellectual material is at the foundation of our civilization's

scientific and technological infrastructure that we now regard as natural as the earth and sky.

While its educational forms evolved into an almost sacred academic tradition (MacLane, 1984) as a

capstone course for which much of the traditional quantitative curriculum could be regarded as

preparation, the ambient societies, the nature of education, and the relations between education and

the larger society, changed and continue to change profoundly.


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More specifically, as the 20th Century came to a close, the received semiotic constraints have been

overcome by the affordances of dynamic and interactive media, and socio-economic conditions

have changed so that now the key ideas underlying calculus must be learned by the great majority

of the population, not merely a technically-oriented elite drawn from a demographically narrow andmainly economically advantaged population. A third profound change, the Type 3 change, is a

shift in the nature of mathematics and science towards the use of computationally intensive iterative

and visual methods that enable entirely new forms of dynamical modeling of nonlinear and

complex systems previously beyond the reach of classical analytic methodsa dramatic

enlargement of the MCV that will continue in the new century (Kaput & Roschelle, 1998).

Despite these profound historic changes, less than 10% of the school population actually completes

the capstone course wherein the key MCV ideas are developed, the curriculum remains organized

around preparation for this course written in the classic algebraic language, most reform efforts,

including calculus reform (see below) continue to take most of these conditions as given, and the

newer MCV is virtually unrepresented in school curricula. Since the institution of education is

deeply connected internally and closely reflects the assumptions and structures of the larger

society, these received givens of content and curriculum define expectations across all aspects of

education: assessment of progress and competence of students, teachers, schools, districts, and

even countries; they define teacher preparation, both pre- and in-service, as well as technology

support of education for learning, for teacher development, for connections between school and

other resources, and so on. These historically rooted expectations, built into the fabric of oursociety and ways of thinking about mathematics science, technology, and education, illustrate the

context and challenge of deep reform.

University Calculus Reform An Illustration of a Type 1 Reform

In the United States these changes, especially technologically-driven changes, and resulting

educational ill-fit with traditional forms, led to a major university-centered "Calculus Reform

Movement" (Tucker, 1990). However, these reforms had two basic characteristics that our current

work, described below, does not share: (1) they were university-centered, intending to reform the

teaching of calculus at the university level without attention to K-12 curricula, and (2) they focused

on the use of interactive technologies to facilitate the learnability and use of traditional notation

systems, both to manipulate within systems as well as to link between representational systems,

especially numeric, graphical and algebraic systems (the traditional "Big Three"). This reform

effort is a good example of an effort that does not employ new representational infrastructures, but

rather improves use and learnability of the inherited ones. Indeed, almost all functions in school


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mathematics continue to be defined and identified as character-string algebraic objects, especially as

closed form definitions of functionsbuilt into the technology via keyboard hardware and input.

SimCalc Representational Innovations A Type 2 Change

An Overview of SimCalc MathWorldsand Its Representations

In order that the cognitive and design issues are understandable, we will provide an overview of

selected aspects of the computer version of our software environment and how it is used in selected

curricular contexts. Parallel versions are available for hand-held devices as well. Indeed, we

regard the desktop computer versions as a supplement and complement to the more widely

accessible versions on hand-held computers. We expect that more than 90% of users will employ

the hand-held versions.

Visually Constructing Functions : In MathWorlds, by choosing an appropriate icon from thevertical toolbar (shown on the right side of the various screens in the screen shots below) the

student or teacher can easily construct a function by concatenating line segments. These can

represent rates of change, such as velocity or acceleration (rate of change of velocity) or price, pay

or tax rates, or they can represent total amounts, such as position (total displacement) or total

amount of money spent, earned or paid in taxes, respectively. The domain variable often is taken

to be time, but need not be, as would be the case in, say, a price-per-item rate, where the domain

(independent) variable is number of items. For example, we could make a step-wise varying

velocity function, where the function appears as discrete steps (constant velocity) as in Figure 1.

We could also make up a velocity function using linearly changing segments (constant acceleration

segments). The point is that the functions need not be described algebraically. Indeed, many of the

functions we create are used to describe situations that would be very difficult to describe

algebraically. In addition, however, MathWorlds can accept standard input of most standard

algebraically defined and hence globally defined exponential and periodic functions, as well as

direct drag-based graphical editing of such functions.

Visually Enacted Actions On Functions : One of the great powers of traditional algebraic-like

mathematical notations is their support for syntactically coherent actions on the notationsrepresenting the functions. One can change their form, compare one with another, combine them,

and so on. This was the extraordinary leap that moved mathematics forward in an entirely new

way in the 16th and 17th centuries (Bochner, 1966). MathWorlds provides a visual analog of

certain actions on functions via direct click-and-drag editing of any segment. For example, a user

can drag the top of a rectangular velocity segment as in Fig. 1 higher to make a faster velocity. Or a


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user can drag the right edge of the rectangular segment to the right to give the motion a longer

duration. Students can also construct a function (or extend an existing one) by adding more

segments to the graph. Thus operations on the representation have clear and simple qualitative

interpretations. For example, Fig. 1 shows a velocity graph that controls the elevator on the left

side of the screen, which will travel at 3 floors/sec for 2 seconds. As indicated in Fig. 4 a linear orpiecewise parabolic position function can be constructed using a single piecewise linear velocity

segment (where, say, a velocity segment can have zero slope, yielding a linear position graph). In

this case, Baby Ducky is controlled by linear position segments (constant velocity) and Momma

Ducky is controlled by parabolic position segments (linear velocity).

Functions Defined by Sampled Data : MathWorlds provides a range of other function types to

complement piecewise or algebraically defined functions. A sampled function type supports

continuously varying positions, velocities, or accelerations. These data points can be entered

directly with the mouse (by sketching the desired curve, ala Stroup, 1996), from Microcomputer-based Laboratory (MBL) data collection gear (Mokros & Tinker, 1987; Thornton, 1992), or by

importing mathematical data from another software package. Motion can also be controlled in real-

time through the use of a mouse-driven "velocity-meter" or "accelerator-meter." A typical scenario

is pictured in Fig. 3, where one vehicle has its motion given in advance and the second vehicle is

controlled by one of the meters in real time. The task might involve following behind the given

vehicle at a specified distance, for example. Furthermore, the given motion might be described via

a position vs. time graph while the student's feedback on the car that she is controlling might be in

terms of a velocity vs. time graph. Here, in Fig. 3 by using the controller on the left to drive the

"VW Bug" with a concave up velocity graph, the student is enacting a typically confusing situation

involving two cars that begin side-by-side but where one has a concave up velocity graph and the

other is to have a concave down velocity graph which crosses the first at a certain point in time.

Well-documented student expectations assume that the cars will be adjacent when their velocity

graphs are adjacent. By "driving" in such situations and many variations on them, the students

come to see not only that this adjacency is not the case, but could never be the case. Fig. 5

illustrates how a sampled function from a motion sensor can drive an actor in a simulationthe

"Froggie Dude" character in the bottom of the picture. A student has created a motion physically

by moving in front of the motion-sensor, an MBL activity. This data has been uploaded toMathWorlds, and attached to Froggie Dude. Then the student created a series of "Clown"

characters and synthetic motions for each using piecewise linear functions. In effect, the student is

"leading his own Clown Parade." Note that Fig. 5 shows the parade in progress, so only the first

part of the graphs is revealed. For orientation to the different kinds of data and notation


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connections possible, see Figure 8, where some notations and phenomena are identified as

Inside the computer, and others as Outside.

Functions Bidirectionally Linked to Phenomena : Throughout, functions drive motions and other

phenomena. And, the other way around. Historically, mathematics has been used to modelsituations that are apart from the mathematics, where processes of abstraction and idealization are

used to mathematize the situation, usually in an iterative way. Simulations provide immediate and

controllable connections between the mathematics and cybernetically defined phenomena. The

ability to import physical data and integrate the data into simulations tightens by orders of

magnitude the connection between the mathematics and the phenomena, both experientially and

temporally. Indeed, the time for feedback cycles of phenomenon-adjustment and mathematics

adjustment is decreased by orders of magnitude. And the kinesthetic connections between physical

actions and immediately visible changes in the model simultaneously opens up new channels for

feedback and conceptual change. Shown in Fig. 8 are situations developed by Nemirovsky and

colleagues at TERC that reverse the data-import-enaction sequence whereby a student creates a

function on the computer and this function, in turn, drives a physical device, such as a car on a

track (shown) or a pump filling a tank (Nemirovsky, Kaput & Roschelle, 1998).

Hot Links Between Functions and their Integrals (accumulations) or Derivatives (rates of change) :

These connections, formalized and systematized by Newton and Leibniz, are related by what has

traditionally been called the Fundamental Theorem(s) of Calculus. In effect, they say (roughly)

that if one starts with a rate description of a varying quantity and forms the accumulation of thatquantity (e.g., start with velocity and determine the position), then the rate of change of the

accumulated quantity is the same as the original varying quantity and vice-versa. This

extraordinarily powerful relationship is at the heart of the power of calculus as a mathematical

discipline, as reflected in its title. We have already noted how the traditional curriculum puts

calculus as a capstone course at the end of a series of algebraic prerequisites. The SimCalc Project

begins with this relationship and builds it into activities and our representations from the very

beginning. Hence we built in a link between the two descriptions to serve activities at the outset,

where a construction is first done in, say, the rate mode, and then it is revisited in the totals mode,

or vice-versa. That is, we frequently treat these two descriptions as providing a second opinion

on each major idea, and often put students in the position of controlling one type of graph while

either the computer or another student controls the other type. Therefore, instead of treating

determining derivatives or integrals as two uni-directional processes, we treat the two kinds of

descriptions as a basic relationship. This is possible because of the simultaneous presence and

immediacy of the connection afforded by the two kinds of graphs built into the learning


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environment. Here all the usual relationships explored in calculus courses through the algebraic

medium as procedures (e.g., taking the derivative) that yield products (the derivative function)

that are then graphable and comparable with the original function can be dealt with as a side-by-

side relationship where each is treated as a description (or driver of) the same phenomenon! Hence

one can work with slopes of position graphs whose values are heights of corresponding velocitygraphs and where each drives the same motion. Furthermore, since they are hot-linked and (if we

choose to configure the system to do so), a dragged change to one is immediately and visually

reflected in its counterpart andis immediately reflected in the phenomenon at hand. Hence

dragging a velocity segment up and down changes the slope of the corresponding position graph

up and down, respectively, and the actor in the motion simulation moves faster or slower,

respectively, during that segment when the simulation is run. See Figure 10, where two functions

are available of each type. We often provide a target function in one mode, say aposition function

controlling object A, and the students task is to match or otherwise interact with that given motion

and description by working with the velocity function for object B. For example, B might follow

A, or two actors might need to exchange places in a certain way as illustrated in Figure 10 and

explored further in the following lesson-scenario.

A Lesson-Scenario Clown & Dude Switching Positions & Eventually Dancing (See Fig. 10) :

For concreteness, consider the following, where, in earlier parts of the lesson from which this

piece is taken, the students were involved in creating graphs to move Clown and Dude around,

switching places at constant speed, coming together and then returning to their original positions,

and so on. (Only step-wise constant velocities have been made available here, although otherfunction types could have been.) The Challenge: Clown and Dude are to switch their positions so

that they pass by each other to the left of the midpoint between them and stop at exactly the same

time. First, walk their motions. Now make aposition graph for Clown and a velocity graph for

Dude so that they can do this.. The student needs to construct graphs similar to #1 and #2. We

have also shown the respective corresponding velocity and position graphs, #3 and #4, which can

be revealed and discussed later. Note that the velocity and position graphs are hot-linked, so

changes in the height of a velocity segment are immediately reflected in the slope of the

corresponding position segment, and vice-versa. Importantly, the activity requires interpretations

of positive and negative velocities, and hence signed number arithmetic, as well as the

representation of simultaneous position. Later activities in the lesson involve a story-line where

Dude is patrolling the area (periodic motion) and Clown gets interested in Dude, follows him at a

fixed distance, harasses him, and eventually, they dancewhere the student, of course, is

responsible for making the dance.


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Determining Mean Values : Fig. 2 shows two velocity graphs, each controlling one of the two

elevators (graphs are color-coded to match the elevator that they control). The downward-stepping,

but positive, velocity graph typically leads to a conflict with expectations, because most students

associate it with a downward motion. However, by constructing it and observing the associated

motion (often with many deliberate repetitions and variations), the conflicts lead to new and deeperunderstandings of both graphs and motion. The second graph in Fig. 2 provides constant velocity

and is shown in the midst of being adjusted to satisfy the constraint of "getting to the same floor at

exactly the same time." This amounts to constructing the mean value of, or the average velocity of,

the other elevator which has the variable velocity. This in turn reduces to finding a constant

velocity segment with the same area under it as does the staircase graph. In this case the total area

is 15 and the number of seconds of the "trip" is 5, so the mean value is a whole number, namely,

3. It is possible to configure MathWorlds so that all segment endpoints have whole number

coordinates - this is denoted and experienced as "snap-to-grid" because, as dragging occurs, the

pointer jumps from point to point in the discrete coordinate system. Note that if we had provided 6

steps instead of 5, the constraint of getting to the same floor at exactly the same time (from the

same starting-floor) could not be satisfied with a whole number constant velocity, hence could not

be reached with "snap-to-grid" turned on.

The standard Mean Value Theorem, of course, asserts that if a function is continuous over an

interval, then its mean value will exist and will intersect that function in that interval. But, of

course, the step-wise varying function is notcontinuous, and so the Mean value Theorem

conclusion would fail as it would if 6 steps were used. However, if we had used imported datafrom a students physical motion, as in Figure 6, then her velocity would necessarily equal her

average velocity at one or more times in the interval. We have developed activities involving a

second student walking in parallel whose responsibility is to walk at an estimated average speed of

her partner. Then the differences between same-velocity and same-position begin to become

apparent. Additional activities involve the two students in importing their motion data into the

computer (or calculator) serially and replaying them simultaneously, where the velocity-position

distinction becomes even more apparent due to the availability of the respective velocity and

position graphs alongside the cybernetically replayed motion.

Note how the dual perspectives illustrated in Figure 9 show two different views of the average

value situation. In the left-hand graph, we see the connection as a matter of equal areas under

respective velocity graphs. In the right-hand graph, we see it through position graphs as a matter

of getting to the same place at the same time, one with variable velocity and the other with constant

velocity.


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Putting Phenomena At the Center, Especially Motion : Underlying all the above illustrations and

worth making explicit is the theme of putting phenomena at the center of the enterprise. This is

partially served by the graphical approach to piecewise linear functions, which allows richer

relations with students' experience of motion. Consider the problem of defining a function that

represents the motion of an elevator that will pick up and drop off passengers in a building. While

such a function is very difficult to formulate algebraically, it is relatively easy to directly drag

hotspots on piecewise linear velocity segments to create an appropriate function. Similarly,

defining motion-functions for two characters who are dancing would be extremely cumbersome to

do algebraically - cumbersome for younger students in entirely unproductive ways. (Exercise:

Write out an algebraic description of the position functions driving Momma and Baby Duckies

depicted in Fig. 4). Equally important to drawing upon children's resources is providing

opportunities to make necessary distinctions in places where prior knowledge may be poorly

differentiated. A classic example is the distinction between slowing down and moving downward(between "going down and slowing down") forced by the step-graph in Figure 2 (Moschovich,

1996). More generally, children have great difficulty distinguishing how much from how fast,

(Stroup, 1996).

By combining the above capabilities, an enormous variety of activities is possible, few of which

have been available to students in ordinary classrooms previously. Before turning to their a few

cognitive considerations and curriculum implications, we will summarize the bigger

representational picture, since it is as the heart of all the other issues.

1) AN ELEVATOR AT 3 FLOORS/SEC FOR 2 SECONDS 2) THEMEAN VALUE OF THE STAIRCASE


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3) DRIVING TOYCARS 4) BABY CATCHES UP TO MOMMA

5) MBL DUDE LEADS A CLOWNPARADE 6) MIXING KINESTHETIC EXPERIENCE WITH

SIMULATIONS


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Figure 7 Hot Connection Between Functions and Derivatives or Integrals


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Notational

Target

Inside Outside

"Big Three"&

Rate-Totals

Simulations

Physical Entities

(Devices orpeople)

LBM

MBLor

mouse

mouse

Off-lineNotations

Figure 8 Multiple Connections Between Phenomena and Models

Figure 9 Math Functions Driving Physical Systems (LBM)


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Figure 10 Switching Positions Using Velocity & Position Functions

Summary of SimCalc Representational Changes

We summarize the core web of five representational innovations employed by the SimCalc Project,

all of which require a computational medium for their realization. The fifth, not discussed above,

is mentioned for completeness, but has not been a sustained focus of our work to date.(1)Definition and direct manipulation ofgraphically definedfunctions, especially piecewise-

defined functions, with or without algebraic descriptions. Included is snap-to-grid control,

whereby the allowed values can be constrained as neededto integers, for example, allowing a

new balance between complexity and computational tractability whereby key relationships

traditionally requiring difficult prerequisites can be explored using whole number arithmetic

and simple geometry. This allows sufficient variation to model interesting situations, avoid the

degeneracy of constant rates of change, while postponing (but not ignoring!) the messiness and

conceptual challenges of continuous change.

(2)Direct connections between the above representational innovations and simulations, especiallymotion simulations, to allow immediate construction and execution of a wide variety of

variation phenomena, which puts phenomena at the center of the representation experience,

reflecting the purposes for which traditional representations were designed initially, and

enabling orders of magnitude tightening of the feedback loop between model and phenomenon.


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(3)Direct, hot-linked connections between graphically editable functions and their derivatives orintegrals. Traditionally, connections between descriptions of rates of change (e.g., velocities)

and accumulations (positions) are usually mediated through the algebraic symbol system as

sequential procedures employing derivative and integral formulas but need not be.

(4)Importing physical motion-data via MBL/CBL and reenacting it in simulations, and exportingfunction-generated data to define LBM (Line Becomes Motion), which involves driving

physical phenomena, including cars on tracks, using functions defined via the above methods

as well as algebraically.

(5)We also employ hybrid physical/cybernetic devices embodying dynamical systems, whoseinner workings are visible and open to examination and control with rich feedback, and whose

quantitative behavior is symbolized with real-time graphs generated on a computer screen.

The result of using this array of functionality, particularly in combination and over an extended

period of time, is a qualitative transformation in the mathematical experience of change and

variation. However, short term, in less than a minute, using either rate or totals descriptions of the

quantities involved, or even a mix of them, a student as early as 6 th8th grade can construct and

examine a variety of interesting change phenomena that relate to direct experience of daily

phenomena. And in more extended investigations, newly intimate connections among physical,

linguistic, kinesthetic, cognitive, and symbolic experience become possible.

Importantly, taken together, these are not merely a series of software functionalities and curriculum

activities, but amount to a reconstitution of the key ideas. Hence we are not merely treating theunderlying ideas of calculus in a new way, treating them as the focus of school mathematics

beginning in the early grades and rooting them in children's everyday experience, especially their

kinesthetic experience, but we are reformulating them in an epistemic way. We continue to address

such familiar fundamentals as variable rates of changing quantities, the accumulation of those

quantities, the connections between rates and accumulations, and approximations, but they are

experienced in profoundly different ways, and are related to each other in new ways.

These approaches are not intended to eliminate the need for eventual use of formal notations for

some students, and perhaps some formal notations for all students. Rather, they are intended to

provide a substantial mathematical experience for the 90% of students in the US who do not have

access to the Mathematics of Change & Variation (MCV), including the ideas underlying Calculus,

and provide a conceptual foundation for the 510% of the population who need to learn more

formal Calculus. Finally, these strategies are intended to lead into the mathematics of dynamical

systems and its use in modeling nonlinear phenomena of the sort that is growing dramatically in


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importance in our new century (Cohen & Stewart, 1994; Hall, 1992; Kaput & Roschelle, 1998;

Stewart, 1990).

A Few Cognitive Considerations

We sought to ground the design of learning activities in a thorough understanding of the

experiences, resources, and skills students can bring to the MCV. We initially examined attempts

by the Scholastics to mathematize change before algebra was available (Claggett, 1968; Kaput,

1994), and took into account the large literature on students' difficulties with kinematics

(McDermott, et al., 1987) and graphs (Leinhardt, et al. 1990). Our aim was to build the ideas to

which the more formal algebraic notations conceptually refer, the ideas that they are "about." These

key underlying ideas of rate of change, accumulation, the connections between variable rates and

accumulation, and approximation, all have forms sensible to young students from diverse

populations. We work with students ranging in age from 6 and 7 years to university students.

Following the historical lead and recognizing that the language and metaphors of motion are usedquite generally to describe change and variation, we focused (although not exclusively) on

mathematizing linear motion, particularly by controlling motion simulations in familiar or fanciful

situations: elevators, people walking or dancing, cars, duckies on a pond, boats in a river, space-

vehicles, and so on (see Figure A).

Research at TERC and elsewhere (e.g., the Shell Centre in Nottingham, England during the

1980s) has uncovered the important roles of physical motion in understanding mathematical

representations (Nemirovsky et al., in press; Nemirovsky & Noble, in press). In studying their

own movement, students confront subtle relations among their kinesthetic sense of motion,interpretations of other objects' motions, and graphical, tabular and even algebraic notations.

Our starting criteria were to begin with students' intuitive experience with speed and motion,

minimize computational complexity, and yet maintain sufficient variation to avoid the conceptual

degeneracy of constant velocity and linear functions (Stroup, 1996). These criteria led to extensive

use of piecewise constant velocity functions as shown in Figures (12). Furthermore, we wanted

to support direct graphical manipulation of these velocity functions - after all, defining and

manipulating piecewise constant functions algebraically is a very cumbersome process, and the

vertical arrow in 2 indicates a dragging action to change the height of the velocity graph segment towhich it is attached.

Yet another major source of design consideration supporting piecewise defined functions, is also

based in the work of our colleagues at TERC, who found that children spontaneously engage in

"interval analysis" to understand the graphical behavior of a complex mathematical function.

Without explicit instruction students parse a graph into intervals based on their understanding of


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the events that the graph represents (Nemirovsky, 1994; Monk & Nemirovsky, 1994), where the

intervals correspond to identifiable, separable sub-events. Within this framework students

understand curved pieces of graphs as signifying behaviors of objects or properties of events,

rather than as sets of ordered pairs in a kind of perceptual subitizing of quantifiable events into

naturally occurring, pre-quantitatively understood chunks. They also readily constructed moreflexible and richer schemes as they made sense of increasingly complex situations and constructed

rich mathematical narratives that tell the story of a graph over time (Nemirovsky, 1996). These

well-documented student resources directly influenced our focus on piecewise defined and editable

functions.

Curriculum Integration Issues: Opportunities and Constraints

Using the MCV to Organize, Contextualize and Energize the Traditional Core

Quantitative CurriculumAn additive approach to curriculum change is impossiblethe curriculum is already overstuffed.

Further, in a standards/accountability environment, one cannot take liberties with the content that is

subject to high-stakes assessment. Hence, to complement offerings of alternative curricular

materials (available for those who can afford to take alternative approaches), we have taken a

transformative approach to curriculum integration. The intent is to enhance the learnability of

traditional, but often difficult ideas such as rate, ratio, proportion, variable, slope, linearity,

function, simultaneous equations, average, signed numbers and areas, periodicity, linear change

(and hence quadratic accumulation), interpretation of graphsall of which appear briefly in the

above examples. Our aim has been to organize these ideas in the service of understanding the key

ideas of the MCV that gave rise to them historically. This in turn means that the students are

simultaneously learning the basic ideas underlying calculus: the different kinds of variation,

relations between rates of change of varying quantities and how they accumulate, continuity and

approximation.

Thus, in reference to the Lesson-Scenario above(see also Figure 10), while the students are

making the two characters in the simulation exchange places while crossing to the left of the center,

they are needing to deal with signed (positive and negative) areas, the idea of variable rate,

simultaneity, and, if one character is driven by a velocity graph and the other by a position graph,

they need to coordinate the relationship between the two kinds of descriptionswhich, as noted

earlier, is the idea at the heart of the Fundamental Theorems of Calculus.


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Furthermore, in engaging in activities that mix physical, simulated and imported motion-data, the

students also develop heuristic skills crucial for life and work in the 21st Century: modeling,

simulation, the differences between physically and cybernetically generated data, how assumptions

play out in models and simulations, etc. At the same time, the use of dynamic simulations

contextualizes and energizes these ideas: students are learning the ideas in the context of deepeningtheir understanding of some phenomenon or as they try to design some dynamic event, such as a

dance, a catch-up situation where a car on a ramp meets traffic on a highway, an elevator trip to

satisfy some constraint, and so on.

In addition, we do not limit attention to the Mathematics of Motion because most of the MCV ideas

apply much more generally, and indeed, apply even more naturally to piecewise defined functions:

consider all sorts of rates with naturally occurring steps, such as tax rates, phone rates, royalty

rates, etc. We expect that the representationally enabled curricular innovations will gradually

infiltrate the mainstream in the next decade. For the newer MCV involving system dynamics, etc.,

rather than curriculum design, we have concentrated on understanding what students bring to our

dynamical systems exemplars, what kinds of knowledge, representations, and actions are needed

to make sense of such systems, and how that knowledge itself is transformed by experiences with

such systems.

Teacher Learning of Content and Understanding of CurricularChanges

We have begun to uncover commonalties and subtleties of teacher learning, having done dozens of

both pre- and in-service workshops for hundreds of teachers of lengths ranging from 2 hours to 25

hours for teachers ranging from elementary teachers to college level teacher-educators. We have

seen that the complexities of the MCV include the needs for deeper understanding of concepts such

as rate and ratio (especially middle school teachers, whose understanding tends to be very

superficial and formulaic), greater fluency with the range of representational media now possible,

and concomitant understanding of the links between and among notations and phenomena (Bowers

& Doerr, 1998). In addition to understanding mathematical interactions and experiences that were

by and large not part of their own mathematical education, teachers also need to understand

learners' conceptual development and hot to create the alternative pedagogical strategies that exploit

our tools (Doerr, & Bowers, 1999). For example, to build concepts of rate we have developed

sequences of activities directed towards both teachers and students using our ability to provide

discretized traces of motion (moving objects drop marks for any specified step-time) (Nickerson,

& Bowers, 1999). These can also become the bases for reformulating approaches to algebra,


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especially linear functions and interpreting slope as rate of change. See especially Nickerson, et

al.(2000).

Reflecting the historical dependence on character strings described at the outset, among high school

and college level teachers, we sometimes see a reluctance to treat our materials as mathematicallyseriousin particular, the unfamiliar graphical mathematics of piecewise constant velocity

functions, and their two-way connections with polygonal position graphs, is seen as secondary to

derivative and integral formulas that apply to globally defined algebraic functions, which embody

the real math. Since the fundamentally graphical approach to the MCV is usually unfamiliar to

teachers, we continue to design activity sequences for teachers that build new understandings about

the relationship between the derivative and the integral. For example, asking such questions as

why does a vertical translation in the velocity graph change the position graph, but not

conversely? often reveals a new insight into the +C of the familiar integration formulas. Most

teachers come to realize that there is much more to this mathematics than derivative and integral

formulas, just as slope is much more than rise over run. Indeed, this mathematics is what the

formula mathematics is about.

The New Issue of Multiple & Networked Hardware & SoftwarePlatforms in Classrooms

Integration of Hand-helds and Larger Computers

Given the rapidly evolving universe of hand-helds and networks, any plans for technology use in

teacher education need to examine how to engage prospective and in-service teachers in optimizingsynergy between hand-held and larger computers, especially where each student has access to a

hand-held device capable of running some version of parallel computer software. Teacher s need

to be able to utilize a desktop or laptop with classroom display capable of running such software as

Java MathWorlds in conjunction with a version running on a popular platform such as the TI-83+,

and where classroom connectivity could range from currently available TI GraphLink 1-1 data

passing between any 2 devices to a full wireless classroom network, and where the hand-held

varies from the decidedly lo-tech but almost universal TI-83+ to wirelessly networked Palm-like

devices.

We have developed a full, document-oriented Flash ROM software system for the TI-83+ and a

core set of activities embodying the curriculum ideas described above that parallels the computer

software to the extent possible given the processing and screen constraints (96 by 64 pixels!) .

The parallelism is evident in the Calculator MathWorlds screens below in Figure 11. (We have

also developed a prototype version of MathWorlds for a PalmPilot.)


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Most user interaction is through the softkeys that appear across the bottom of the screen which are

controlled by the hardkeys immediately beneath them. The left-most screen depicts the Animation

mode, with two elevators on the left controlled respectively by the staircase and constant velocity

functions to their right. The right-most screen shows a horizontal motion world with both position

and velocity functions displayed (hot-linked, as with the computer software). The middle screen

depicts the Function-Edit mode, which shows a hot-spot on the constant-velocity graph. The

user adjusts the height and extent of a graph segment via the 4 calculator cursor keys (not shown),

and can add or delete segments via the softkeys. Other features allow the user to scale the graph

and animation views, display labels, enter functions in text-input mode, generate time-position

output data, and so onvery much in parallel with Java MathWorlds, but without the benefits of a

direct-manipulation interface.

Figure 11 MathWorlds for the TI-83+

Studies of Classroom Interactions

We now ask a critical question:How many of our activity-snippets above can be done in this

environment? The answer is almost all of them. Indeed, our core MCV curricula for pre-

algebra, algebra and precalculus can be executed with this system. Another question: Why

sacrifice all the power and visual capability of computers? The small device supports only 2

objects, limited scale, and only schematic one-dimensional motion worlds, and the computer

software supports motions along user-defined paths as well as 2-dimensional change enabling

richer and more complex activities. But hand-helds offer continual classroom availability, low cost

(about a 5th the cost of a computer lab to equip a class including one computer and display) and

portability. Hence a rich activity introduced on the teachers computer/display can be followed-up

by individual or small-group activity, including homework, on the hand-helds.

Increasingly rich interactions are possible as connectivity increases between a teachers computer

and a classroom of hand-helds. For example, a teacher can download sets of documents for

homework or quizzes, and more interestingly, the students can upload their solution-documents as


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well as other data, which can then be aggregated in a variety of ways on the teachers computer.

For example, groups of students can act out or choreograph a collective motion, say a dance,

collectively, and then sit down to plan the coordination of their individual motions as mathematical

functions that they will produce on their hand-held. They then upload their individual functions to

the teachers computer where the serially produced motions are aggregated into a simultaneouslyexecuted dance to be viewed by the entire class! This amounts to a netwroked version of the

Marching Parade activity depicted in Figure 5. Variations of this kind of aggregation activity can

use CBL input as well, and a wide variety of other aggregation and target activities is possible.

In early prototype testing , we found subtle perceptual carryovers from the computer to the

calculator environments that may provide guidance on how to exploit the visual detail possible on

the computer screen to compensate for limited screens of hand-helds. For example, despite the

hard to read grid of the calculator screen, the students, who were often presented activities using

graph printouts based on the computer screens, seemed to treat the calculator screen as havingvisual attributes that were present only on the computer software. These kinds of potentially

important phenomena need to be studied and documented in more detail, as do potential

interference effects across the different environments.

We are currently pursuing research with several private sector partners, including Texas

Instruments and Palm, to examine the affordances and constraints of networked mathematics

classrooms employing mixes of hardware and software platforms. Of particular concern are issues

of implementability and teacher knowledge, content knowledge as well as pedagogical knowledge

and how these interact with the various technological options available. These results will have adirect bearing upon the design of pre- and in-service experiences for teachers.

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