8/3/2019 Kaput. Representacin, Original.
1/24
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.
8/3/2019 Kaput. Representacin, Original.
2/24
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,
8/3/2019 Kaput. Representacin, Original.
3/24
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
8/3/2019 Kaput. Representacin, Original.
4/24
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.
8/3/2019 Kaput. Representacin, Original.
5/24
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
8/3/2019 Kaput. Representacin, Original.
6/24
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
8/3/2019 Kaput. Representacin, Original.
7/24
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
8/3/2019 Kaput. Representacin, Original.
8/24
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
8/3/2019 Kaput. Representacin, Original.
9/24
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.
8/3/2019 Kaput. Representacin, Original.
10/24
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.
8/3/2019 Kaput. Representacin, Original.
11/24
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
8/3/2019 Kaput. Representacin, Original.
12/24
3) DRIVING TOYCARS 4) BABY CATCHES UP TO MOMMA
5) MBL DUDE LEADS A CLOWNPARADE 6) MIXING KINESTHETIC EXPERIENCE WITH
SIMULATIONS
8/3/2019 Kaput. Representacin, Original.
13/24
Figure 7 Hot Connection Between Functions and Derivatives or Integrals
8/3/2019 Kaput. Representacin, Original.
14/24
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)
8/3/2019 Kaput. Representacin, Original.
15/24
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.
8/3/2019 Kaput. Representacin, Original.
16/24
(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
8/3/2019 Kaput. Representacin, Original.
17/24
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
8/3/2019 Kaput. Representacin, Original.
18/24
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.
8/3/2019 Kaput. Representacin, Original.
19/24
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,
8/3/2019 Kaput. Representacin, Original.
20/24
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.)
8/3/2019 Kaput. Representacin, Original.
21/24
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
8/3/2019 Kaput. Representacin, Original.
22/24
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.
References
Various versions ofMathWorlds for computers and calculators , along with other articles and
materials, can be downloaded from the SimCalc web site, http://www.simcalc.umassd.edu
Bochner, S. (1966). The role of mathematics in the rise of science. Princeton, NJ: PrincetonUniversity Press.
Bowers, J. S. & Doerr, H. M. (1998, October). Investigating teachers insights into themathematics of change. In S. Berenson, K. Dawkins, M. Blanton, W. Coulombe, J. Kolb,K. Norwood, & L. Stiff (Eds.), Proceedings of the Twentieth Annual Meeting of the NorthAmerican Chapter of the International Group for the Psychology of Mathematics Education(Vol. 2, pp. 789-795), North Carolina: North Carolina State University.
Boyer, C. (1959). The history of calculus and its historical development. New York: DoverPublications.
8/3/2019 Kaput. Representacin, Original.
23/24
Claggett, M. (1968). Nicole Oresme and the medieval geometry of qualities and motions,Chapters I & II. Madison, WI: University of Wisconsin Press.
Cohen, J. & Stewart, I. (1994). The collapse of chaos: Discovering simplicity in a complexworld. New York: Viking Books.
Confrey, J. (1991). Function Probe [software]. Ithaca, NY: Department of Education, CornellUniversity.
Doer r, H . M. & Bow ers, J. S. (1999). Revealing pre-service teachers th inking abou tfunctions thr ough concept m ap ping. In F. Hitt & M. Santos (Ed s.), Proceedings of
the Tw ent y First A nn ual M eeting of the N orth A merican Chapter of the
Int ernational Group for the Psychology of Education , (Vol. 1, pp. 364-369).Duckworth, E. (1991). Twenty-four, forty-two, and I love you: Keeping it complex. Harvard
Educational Review, 61(1), 124.Edwards, C. (1979). The historical development of the calculus. New York: Springer-Verlag,
Inc.Hall, N. (1992). Exploring chaos: A guide to the new science of disorder. New York: Norton.Kaput, J. (1994). Democratizing access to calculus: New routes using old roots. In A.
Schoenfeld, (Ed.),Mathematical thinking and problem solving. Hillsdale, NJ: Erlbaum.Kaput, J., & Roschelle, J. (1998). The mathematics of change and variation from a millennial
perspective: New content, new context. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.),Mathematics for a new millennium (pp. 155170). London: Springer-Verlag.Kaput, J. & Roschelle, J. (1996). Connecting the connectivity and the component revolutions to
deep curriculum reform (http://www.ed.gov/Technology/Futures/kaput.html). Washington,DC: Department of Education.
Kaput, J. & Shaffer, D. (in press). On the development of human representational competencefrom an evolutionary point of view: From episodic to virtual culture. In K. Gravemeijer, R.Lehrer, B.van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use inmathematics education. London: Kluwer.
Leinhardt, G., Zaslavsky, O., & Stein, M. (1990). Functions, graphs, and graphing: Tasks,learning, and teaching. Review of Educational Research , 60, 164.
MacLane, S. (1984). Calculus is a discipline. The College Mathematics Journal, 15(5), 375.Mahoney, M. (1980). The beginnings of algebraic thought in the seventeenth century. In S.
Gankroger (Ed.),Descartes: Philosophy, mathematics and physics. Sussex, England:Harvester Press.
McDermott, L., Rosenquist, M., & Zee, E. (1987). Student difficulties in connecting graphs andphysics: Examples from kinematics. American Journal of Physics, 55, 503513.
Mokros, J.R. & Tinker, R.F. (1987). The impact of microcomputer-based labs on childrensability to interpret graphs. Journal of research in science teaching, 24(4), 369383.
Monk, S.; Nemirovsky, R. (1994). The case of Dan: Student construction of a functionalsituation through visual attributes. In E. Dubinsky, J. Kaput & A. Schoenfeld (Eds.),Research in collegiate mathematics education (Vol. 4, pp. 139168. Providence, RI: AmericanMathematical Society
Moschovich, J. N. (1996). Moving up and getting steeper: Negotiating shared descriptions oflinear graphs. Journal of the Learning Sciences, 5(3), 239277.
Nemirovsky, R. (1996). Mathematical narratives. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
Approaches to algebra: Perspectives for research and teaching (pp. 197223). Dordrecht, TheNetherlands: Kluwer.
Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and velocity sign. TheJournal of Mathematical Behavior. 13, 389422
Nemirovsky, R., & Noble, T. (in press). Mathematical visualization and the place where we live.Educational Studies of Mathematics.
Nemirovsky, R., Kaput, J. & Roschelle, J. (1998). Enlarging mathematical activity from
modeling phenomena to generating phenomena. Proceedings of the 22nd Psychology of
8/3/2019 Kaput. Representacin, Original.
24/24
Mathematics Education Conference in Stellenbosch, (Vol 3., 287-294). Stellenbosch, South
Africa.Nemirovsky, R., Tierney, C., & Wright, T. (in press). Body motion and graphing. Cognition
and Instruction.
Nickerson , S., Nyd am , C. & Bow ers, J. S. (2000). Linking algebr aic concepts and
contexts: Every picture tells a story.M athematics Teaching in t he M iddleSchool 6(2), 92-98.
N ickerson , S. & Bow ers, J. (1999, October). Inves tigatin g stu dent s d evelop ing
conceptions of rate. In F. H itt & M. Santos (Eds.), Proceedings of the Twenty First
Annual Meeting of the North American Chapter of the International Group for
the Psy chology of Education, (Vol. 1, p. 410).Roschelle, J. (1992). Learning by collaborating: Convergent conceptual change.Journal of the
Learning Sciences, 2(3), 235276.Roschelle, J. & Kaput, J. (1996). Educational software architecture and systemic impact: The
promise of component software. Journal of Educational Computing Research, 14(3),217228.
Roschelle, J., Kaput, J, & Stroup, W. (2000). SimCalc: Accelerating students engagement with
the mathematics of change. In M. Jacobson & R. Kozma (Eds.)Educational technology andmathematics and science for the 21st century (pp. 4775). Hillsdale, NJ: Erlbaum.
Stewart, I. (1990). Change. In L. Steen (Ed.), On the shoulders of giants: New approaches tonumeracy (pp. 183219). Washington, DC: National Academy Press.
Stroup, W. (1996). Embodying a nominalist constructivism: Making graphical sense of learningthe calculus of how much and how fast. Unpublished doctoral dissertation. Cambridge, MA:Harvard University.
Thornton, R. (1992). Enhancing and evaluating students' learning of motion concepts. In A.Tiberghien & H. Mandl (Eds.), Physics and learning environments [NATO Science Series].New York: Springer-Verlag.
Tucker, T. (1990). Priming the calculus pump: Innovations and resources. Washington, DC:MAA.