Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
Monograph XIII, 2005
Chapter 3:
Epistemological
and Cognitive
Aspects of Time:
A Tool Perspective
Michal Yerushalmy,
University of Haifa
Beba Shternberg, Center for
Educational Technology
Abstract
In this study, we focus on the first steps of
qualitative modeling of temporal phenomena. In particular, we analyze subtleties involved in the analysis of images of
“hand-made motion.” We start with the
description of the rationale of the environment on which this study is based. We
then describe two pairs of seventh graders
who are involved in modeling a given
physical phenomenon. Episode 1 (approximately one minute) involves the efforts
of one pair of students to understand the
correlation between graphs and the path
produced by the motion of a hand. We
attempt to identify various issues students
encounter in explaining how the trajectory
of the hand is represented in graphs of
“x over time” and “y over time.” Episode
2 (approximately two minutes) concerns
a dialogue between two students about
the role and representation of time in the
graphs. Their discussion brings up issues
regarding time as an independent variable and as a causal influence on the
graph. We close by discussing connections among different fields of modeling
as reflected in the design of the learning
environment engaged in this study.
Introduction
“Then why do we have time here?”
“Time is always there. If there were no
time …, it would not have moved.”
(Hila and Alon, seventh graders)
How do students construct the meaning
of a mathematical representation of time
when using tools?
This paper focuses on two episodes with
seventh-grade students modeling temporal phenomena by mathematical symbols
for the first time, using a motion-construction software tool, referred to here as
the “tool.” Play [Segment A] to observe
an introduction to the software tool. The
students used the tool to learn to create
models of motion (i.e., graphs in a Cartesian plane) and to construct the meaning
of mathematical symbols by manipulating
the symbols and reasoning about them.
Throughout a tool-perspective analysis,
we wanted to analyze understanding of
the effect of a certain action with the tool
(e.g., holding down the mouse while the
system was running) and its mathematical
meaning (time as an independent variable)
as they emerged in a simultaneous and
coordinated manner.
The purposes of this paper are to (a)
analyze the way in which an understanding of the mathematical symbols of time
emerges, (b) discuss a central component
of understanding—a type of abstraction situated in the tool, and (c) analyze
the connection between the emergence
of understanding of the mathematical
presentation of time and the development
of the perception of time as elaborated by
Piaget.
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
The Tool Perspective of Study
Why a Tool Perspective?
The episodes presented here are analyzed
from a tool perspective. In technologybased environments designed to support
learners in expressing themselves mathematically and manipulating mathematical representations (Yerushalmy, 1999;
Yerushalmy & Schwartz, 1999; Yerushalmy
& Shternberg, 2001), the tools provide a
set of terms and mathematical symbols;
thus, the mathematical use of the tool
and reasoning with it are carried out in the
terms and symbols designed into the tool.
Activities that use specifically designed
technology also shape the language,
discourse, and construction of meaning produced in the course of the activity
(Noss, Hoyles, & Pozzi, 2002). The tool
plays more than one role here. First, the
terms designed into the tool affect the way
in which students construct the meanings of both the symbols of the tool itself
and the concept the activity is intended
to explain. At the same time, students
can use the symbols embedded in the
tool in many different ways, so a different
construct of meaning emerges each time.
Analysis from the tool perspective is thus
challenging because the knowledge and
the meaning of concepts that students
construct with the tool are often different
from those they construct by other means.
When the meaning of concepts and
symbols emerges in the context of a tool,
the analysis of the cognitive development
often incorporates specific characteristics
that originate in the interaction with the
tool. Analyzing these instances can lead
to a reconsideration of the terms used to
describe them (Artigue, 2002; Cobb, 2002;
Nemirovsky, Tierney, & Wright, 1998).
In the context of this study, analysis from
a tool perspective refers to (a) the various
uses of the tool in different parts of the
study; (b) the emergence of the meaning
of time as reflected by direct manipulation of symbols within the tool; and (c)
students’ mathematical reasoning using
the tool.
Tool-Situated Abstraction
Within the complex activity that is learning, technology can create situations that
are suited for the study of mathematical
concepts. Analyzing the way in which
mathematical knowledge and abstractions
are constructed and represented with the
aid of tools is a central issue in what Noss
(2001) defines as digital culture. Conventional mathematical concepts, ideas, and
abstractions are often presented in somewhat nonconventional ways, depending on
the nature of the tool.
Several attempts have been made to find
suitable terms for describing the situational emergence of meaning. Nathan (1998)
analyzes feedback as situated; Noss et
al. (2002) discuss situated abstraction by
hospital nurses; Nemirovsky (2002) distinguishes formal from situational generalization. Common to all discussions about
situated learning is the assumption that
a situated knowledge of concepts is not
detached from the circumstances of the
situation. But regarding the construction
of mathematical knowledge, pure abstraction, which usually requires detachment
from the objects and events of the situation, is widely accepted as an essential
component of the mathematical learning
process. The necessary conclusion seems
to be that the construction of mathematical knowledge in situated learning cannot
include abstraction and therefore cannot
be complete.
Abstraction is an entity that presents the
essential features of objects and events
and eliminates inessential properties; it
generalizes all relevant elements and only
those elements, and therefore appears
incongruous in the circumstances of a
specific situation. Another way of looking
at abstraction is as a process. The Oxford
English Dictionary defines abstraction as
“the act of separating in thought.” Hershkowitz, Schwarz, & Dreyfus (2001) suggest
distinguishing between the process of abstraction and its outcome using different
terms: abstracting and abstracted entity,
respectively. Herscovics (1996) describes
processes of abstraction as “involving
two phases: the first one consisting of
the separation of the concept from the
procedure [that constructs the concept],
and a second phase characterized by the
generalization of the concept, or by some
form of conservation reflecting the invariance of the mathematical object, or by the
reversibility and possible composition of
the mathematical information” (p. 359).
In Herscovics’s (1996) view, therefore, an
abstraction need not be detached from
the circumstances of its situation as long
as it involves a process of disassociation
from particularities. This view describes
better the process of abstraction within
a tool, which refers to the construction
of “new things” or “new usages,” mental
or physical, that exist inside the tool but
express an abstract entity or a process of
abstraction as a way of describing an idea
detached from some particularities.
This paper analyzes episodes looking for
abstraction that takes place with the tool.
This type of abstraction is referred as tool
situated abstraction.
One of the aims of the study is to indicate
processes that occur with the tool and
could be viewed as abstraction.
The Tool Used in the Study
The tool used in this study is part of the
Function Sketcher motion-construction software (Yerushalmy & Shternberg,
1999/1993) and is a component in an
environment designed to support the first
stages of qualitative modeling as a prealgebra activity. The following description is
restricted to components relevant to the
episodes under study.
The tool supports the interaction between
the mechanical motion of a computer
mouse and the formulation of a graphical model, allowing students to create or
simulate planar motion using freehand
mouse drawing. The planar movement of
the hand provides the input that appears
on the screen as a graph of a two-dimensional path.
The tool stores x-y position values correlated to the duration of motion. The quantities involved in motion do not appear on
the initial planar graph, but they are shown
as they vary over time on data graphs that
appear simultaneously or on request in
two separate windows. The student uses
a clock icon to indicate the beginning and
end of the experiment. Figure 3.1 shows
the path drawn on the x-y plane (top left)
and two graphs representing the x- and
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
y-coordinates against time (bottom and
top right).
Figure 3.1. Planar graph and x and y position graphs.
The planar graph alone shows the trajectory of motion but does not suggest
succession or duration of motion. The tool
as a whole, however, makes explicit the
coordination of the spatial and temporal
characteristics of the motion with the aid
of the x-in-time, or x(t), and y-in-time,
or y(t), graphs. As shown in Figure 3.2,
the same planar graph can represent a
trajectory of completely different motions
in time.
Figure 3.2. Two different sets of graphs produce identical paths.
All graphs are drawn on a Cartesian plane.
The Cartesian plane is used as a locator, in
which neither x nor y has a privileged role,
alongside two other Cartesian graphs,
each representing a locator, x or y, as a
function of time. This multiplicity of views
of the Cartesian plane introduces a challenge of moving flexibly among the views.
A look at philosophical investigations of
the meaning of time and at Piaget’s views
of understanding the concept will help
in understanding the emergence of the
meaning of time and of its symbols by
interacting with the tool.
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
Meanings of Time
The Meaning of Symbols
The connection between the meaning of a
concept and that of its mathematical symbol is not always obvious. Various notions
of the meaning of symbols have been
studied in mathematics education, psychology, and philosophy (e.g., Foucault,
1966; Vygotsky, 1997). This study regards
a symbol as a signifier that represents,
signifies, or replaces something (Foucault,
1966; Luria, 1976). Each symbol contains
two ideas: that of the signifier and that
of the signified. Developing the meaning
of a symbol is a compound process of
conjectures, analyses, and descriptions of
the tenor, in this case, the concept that the
symbol might represent. The process of
constructing the meanings of a symbol is
connected with the process of constructing the meanings of the concept, and both
are viewed in the context of the language
used to describe them.
Studying the development of the meaning of symbols has strong implications for
the study of understanding. An operative
method to study meaning was suggested
by Wittgenstein (1973), who proposed that
the meaning of signs should be understood by the way in which they are used
within their context.
The Meaning of Time
Constructing meaning is a complex
process when it requires coping with the
mathematical representation of some
component of a real phenomenon. As the
vagueness of the signified increases, so
does the complexity of the task. Time is a
case in point. The concept of time, and of
time as a variable in models, is a mental
construct. Involving other concepts that
can be used as reference to construct the
meaning of such a concept is necessary;
one can observe motion, for example, to
learn the meaning of time.
Philosophers look for proper ways to talk
about time and to answer the question of
whether time is a component of temporal
phenomena in reality. The answers vary.
According to Plato, time “is the circular
motion of the heavens.” Aristotle treats it
as the measure of motion. Saint Augus-
tine, in a view somewhat similar to later
theories of Kant, regarded time merely
as a form of sensible intuition. In Kant’s
view, space and time are conditions of the
existence of things as phenomena and
not elements for the cognition of things.
A more recent definition treats time as the
dimension of causality of events (Reichenbach, 1924/1959). In this paper, time
is related to external physical temporal
processes, such as motion.
Piaget (1974) took a stand against Kant’s
notion that our concepts of space and
time are innate: “He [Kant] was wrong
to conclude that time and space were
‘forms of sensibility,’ and hence to deny
their operational character. In fact, space
and time result from operations just as do
concepts (classes and logical relations)
and numbers” (p. 34). Piaget identified
a relation between our understanding of
time and our grasp of space and causality. According to him, constructs whose
build-up begins very early are present, in
the course of what is called circular reactions, during the first 2 years of an infant’s
cognitive development. He suggests the
concepts of space and time arise in conjunction with those of object permanence
and causality. In the very first sentence of
The Child’s Conception of Time, Piaget
claims that the concept of time has no
meaning outside of a kinetic context.
Specifically, he writes, “Space is a still of
time, while time is space in motion” (p. 2).
Consistent with the modern definition of
time in philosophy, Piaget argues that time
is inherent in causality. To determinate
time, one must determine the order and
duration of successive events, explaining
the later in terms of the earlier: “Since the
motion itself serves as a clock, and since
its duration is judged by its terminal point
in space, temporal succession must be
‘centered’ on spatial succession” (p. 119).
However, according to Piaget, the perception of motion, temporal order, spatial
order, and duration and of the links among
them go through a cognitive development.
“Motion is originally conceived of [by a
child] as a function of the final position of
the moving body, in other words as a function of the goal to be reached” (p. 119).
Duration is ignored in the first stages, and
as long as the child ignores duration, dissociation of the temporal from the spatial
order remains intuitive. Children capable
of reconstructing motion intuitively are
doing it by arranging successive events
purely by their spatial characteristics and
can still be unable to treat this succession
as a function of motion. When time is first
constructed, a child does not conceive it
as something unique. Piaget states, “It is
of considerable psychological interest to
discover the precise nature of a conception (the unicity of time) that is lacking at
first, appears quite naturally at a certain
level of intellectual development, and is
then surpassed…” (p. 49). Thus, Piaget
believes that for most adults, time is in
fact an “a priori form of sensibility” (p. 49). Kinds of Time
Despite its nonmaterial nature, or perhaps
because of it, time has been classified and
named in many ways, from qualitative and
quantitative through heterogeneous and
homogeneous to inner and absolute time,
and so on (for an extensive review, see
Brann, 1999).
To investigate the process whereby the
meaning of time emerges through the
symbols used by the tool, one must distinguish between physical and psychological
time. Human time, in general, and “now,”
in particular, are defined not in terms of a
physical clock but rather of events. The
sentence “I am studying now” may refer to
a specific moment or to a period in one’s
life.
Physical time corresponds to the human
need for orientation in physical space and
is a component of quantitative relations
between spatial succession and external
events; psychological time is best understood as a consciousness of physical
time. Physical time is public time; psychological time is private time. One refers
to physical time when one speaks of the
time measured by the clock or defines
speed as the rate of change of a position
with respect to clock time; one refers to
psychological time when one says that
“time flies when you’re having fun.” Physical time never stops, is always present,
and is always “now time” because past
and future do not exist. Psychological time
stops when consciousness does; past and
future exist in psychological time.
According to Piaget, during the first
phases of representative thought, children
do not succeed in estimating concrete
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
duration or rates of speed except by referring to psychological time. The children
described in the present study are 13
years old, but Piaget’s distinctions about
younger children may help in interpreting
the episodes presented here.
The Setting
This study is based on an episode involving two pairs of students, Alon working
with Aric and Maya with Hila. They are
in the seventh grade and are ages 13 to
14. The episode focuses on Alon and
Hila. They were in their first year in junior
high school, and they were beginning to
explore modeling temporal phenomena
as a prealgebra activity. The episode is
divided in two parts. It occurred during
regular algebra class in an urban school,
in which teachers have been experimenting for a few years with the VisualMath
curriculum, a functional approach to
algebra described briefly by Yerushalmy
and Shternberg (2001). The students had
had some experience plotting and reading
points on a Cartesian plane but no formal
experience with variables. The episode
occurred during one of the first activities
designed to support nonsymbolic modeling of real-world phenomena.
To base the discussion on familiar temporal processes, students were asked
to bring to class authentic situations
from their experience (e.g., stories from
newspapers) and to describe them in
mathematical narratives. They had never
before used the tool, nor were they familiar
with the specific activity investigated here.
The class was composed of approximately
30 students, working in pairs, one pair to a
computer. Before the interaction between
Alon and Hila began, Alon was paired with
Aric, and Hila was paired with Maya.
As an introductory activity to motion
construction with the tool, students spent
some time drawing a path without motion.
The tool clock was not activated, and
students were getting acquainted with
moving the mouse and observing the path
in the x-y plane on the screen. Later they
started the clock, which recorded x-y locations in time while the paths were being
drawn and redrawn, and generated the
graphs of x and y positions in time. Students shared their findings with their partners and were getting ready for a discussion in a larger group with their teacher.
(They had been participating in the algebra
course for a few months, and the procedures had already become routine.) When
the students were well acquainted with
the software, they were asked to play a
new game called “Hava’s ball,” which was
described to them as follows:
“Hava loves to play ball. When she throws
it to the ground, the ball hits the ground
and then the wall. Sketch the ball’s path,
and describe what the x-time and y-time
graphs tell you.”
“Hava’s ball” was the first in a sequence
of activities designed to help students
experience quantities that change in
time continuously. The activity provides
an opportunity to observe how everyday
physical phenomena turn into geometrical
figures and graphs that are mathematical
models. The tool perspective analysis of
“Hava’s ball” focuses on these questions:
1. How is the tool used to mediate
between the image of a given
physical phenomenon and its
mathematical representations?
2. How is the tool used to construct a “tool phenomenon”?
3. How do reproduction and interpretation of phenomena develop
using mathematical symbols?
4. What kinds of time emerge
from interpreting the x(t)
and y(t) graphs?
5. What are the indications of
abstraction situated in the tool?
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
Episode 1:
Touch the floor, hit the wall
ger than the time spent on the way back;
the point where the ball strikes the wall is
not at the same height as Hava’s hand;
and so on.
Drawing the Picture
Drawing x(t)
Both pairs of students began by drawing
the path followed by the ball. Alon and
Aric retold the story as they were drawing
the graph (Aric had control of the mouse):
The episode continues with Alon describing the x(t) graph, that is, the change in the
horizontal distance from Hava to the wall
and back. See Figure 3.5.
*
Alon: In effect, we are drawing
“Hava,” what happens to her ball.
*
Aric: Now we go up.
Here is Hava.
*
Alon: It touches the floor, then it
hits the wall, goes back down,
and comes back to Hava.
From the outset, Alon seemed aware of
the fact that their activity was going to be
somewhat artificial: “In effect we are drawing ‘Hava.’” Judging by the mouse path
they drew, their conversation, and the behavior of the students during the episode,
they were concerned neither with physical
authenticity (i.e., gravitation, velocity, acceleration) nor with geometrical accuracy.
When drawn accurately, the path produces two halves of a parabola. Alon and Aric
drew it as four curved segments. The two
points representing the moments when the
ball touched the ground, going and coming, were close but did not overlap; this
representation seems close to the actual
experience, but they did not raise the
issue at all. The students described the
process mainly by its four events: touching the ground, touching the wall, touching
the ground again, and returning to Hava’s
hands. They treated the computer screen
and the axes as objects in the perpendicular plane in which the ball traveled: they
chose the y-axis to be Hava, the x-axis
to be the floor, and the right edge of the
window to be the wall.
While they were drawing the path, the
graph showing the y position of the path
versus time appeared in a window to their
left. See Figure 3.3.
Figure 3.3. Drawing and graph.
The students paid no attention at all to the
left side of the screen. They concentrated
on verbalizing the situation as Alon retold
the story, pointing at the drawing with the
mouse.
Drawing y(t)
Next they turned their attention to the y(t)
graph, repeating the narrative and pointing
to the representation of the events in the
graph (Figure 3.4).
4. Alon: Well, they say here, “Hava
likes to play with the ball. She
throws it to the ground. The ball
hits the ground and then it hits
the wall.” So, here it hits the
wall, goes back to the ground,
and then goes back to Hava.
5. Alon: Here is Hava. The
graph says it goes down.
6. Alon: It goes up, it hits the
wall. It goes down. Because
of the wall it falls down and
goes back to Hava’s hands.
Figure 3.4. The y(t) graph.
The y(t) graph also consists of four lines
corresponding chronologically to the four
processes of the physical phenomenon.
The four events were probably easy to
identify on the graph. Touching the graph
with the mouse, they retold the story: “The
graph says it goes down.” They added a
little explanation: “Because of the wall it
falls down.” But the y(t) graph provides
further details of the path, which were left
unexamined by the students. The graph
is not completely symmetrical: The time
spent on the ground on the way up is lon-
7. Alon: The x-coordinate.
It is not important.
8. Alon: Oh, that’s hard to
understand. Hmm.
9. Alon: I think that here it is
Hava’s ball. She threw it to
the ground. Here somewhere
it stayed on the wall. It hit the
wall and stayed here for a while,
just stayed in the middle simply
because of “time,” and then
it bounced back to Hava.
10. Alon: And here it is just an x….
Figure 3.5. The x(t) graph.
When graphed on a vertical axis, the
horizontal distance produces a trace with
a shape that does not resemble the path
of the ball. The graph does not appear to
be made up of four distinct segments, and
the events in time are less transparent.
Glancing at the x(t) graph, Alon immediately identified a complexity: “Oh, that’s
hard to understand.” Not concerned with
the graphical representation of x versus
time, he did not even look at the original planar graph on the right side of the
screen as he described the x(t) graph; the
software allowed only one graph at a time
to appear next to the planar graph, so the
x(t) image replaced the y(t) image. Attempting to link the parts of the x(t) graph
with the story of the ball’s path, Alon retold
the narrative a third time (as did Noam in
Noble & Nemirovsky, 1995). Alon’s ability
to grasp this nontrivial graphic representation as a model of motion is also obvious
in Segment D at points (7) and (10)—the
beginning and the end—where he points
toward the constant function at the end of
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
the motion and comments that “it is not
important.” Indeed, this segment of the
graph does not belong to the story, and it
represents Aric’s slow response at a time
when the process should have ended
already.
Alon’s Answers
Alon took the mouse from Aric, who until
then was in control of it, and answered
Hila while drawing on the Function
Sketcher and watching the x(t) graph (see
Figure 3.6):
13. Alon: It is not exactly …
Episode 2:
Bringing time to the fore
14. Alon: I’m taking it back, and then
it goes down to the negative
part. I’m taking it forward, and
then it goes up to positive part.
This episode concerns the role of time
as an independent variable in Cartesian
graphs. The 2-minute video segment
shows the discussion between Aric and
Alon, on the one hand, and that between
Maya and Hila, on the other.
15. Alon: And if I don’t move it,
“time” will continue to pass.
While Aric and Alon moved on to another
task, Maya and Hila, seated at a computer
next to them, had also been drawing the
ball’s path and watching the y(t) and x(t)
graphs. Glancing over their computer,
Beba heard an interesting conversation
between Maya and Hila. Hila was asked
to clarify to a group of four students what
she and Maya had been discussing.
Hila’s Question
In a group of four, Hila repeated the issue
she and Maya had just started to explore.
Her questions immediately stimulated
Alon.
11. Hila: Why does x depend
on …? What is dependent
on us? Or independent?
12. Hila: For instance, time. It
passes. We can’t stop it. Or,
for example, our growth. We
can’t stop it. These are things
that don’t depend on us.
Hila expressed her dissatisfaction with
their explanation of the graphic images.
Her words lacked clarity, but she expressed a difficulty with time as an independent variable and x as a dependent
variable.
After a short time, Alon joined the discussion.
Figure 3.6. Alon’s actions.
Illustration of the Software
Alon responded to what he thought was
Hila’s first concern: her question about
how the x she had drawn with her own
hand had become a dependent quantity.
Demonstrating that he could set x despite
its being a dependent quantity in a graph
that also showed time, Alon showed he
could control x while drawing the path. He
neutralized the role of y and concentrated
on motion in a single dimension, probably
to demonstrate his control over the position of x. He moved the mouse only to the
left during the first time interval and only to
the right during the second. Thus, he was
able to demonstrate that only he controlled the x path, just as Hila had argued.
He then focused on another part of her
question: Why does x depend on time?
Directing Hila to view the x(t) graph simultaneously with the drawing, he showed
how time starts at zero and increases
constantly. Thus he demonstrated that the
shift in the direction of the x-coordinate
in the planar graph occurs at the same
moment as the change in direction in the
x(t) graph. Alon probably hoped that the
demonstration of the simultaneous events
would convince Hila that x was both controlled by, and dependent on, time.
By designing an experiment with the
Function Sketcher, Alon found a way to
respond to Hila’s argument about time
flowing without control (12). Alon created a situation in which time flows freely.
Disabling not just the vertical variable of
the motion (14) but both variables (15),
Alon held the mouse in place. Although
the x-coordinate did not change, the flow
of time made the x(t) graph continue and
become a constant line. (“If there were no
time, it would have been a point. It would
not have moved.”) His argument that the
input was a null action modeled only by
free-flowing time seemed to be a strong
one.
Pedagogically, Alon used two strategies
with the tool: He supported his interpretation of the correct arguments (Alon also
realized that time flows freely and that the
hand controls the x position, but this idea
did not confuse him); and he tried to help
Hila refute her incorrect interpretation that
x cannot be dependent on something else
(for Alon x(t)-dependence was perfectly
reasonable). He used a specific feature
of the tool to execute these strategies,
showing two images simultaneously. But
more important, he logically sorted out the
potentially confusing variables, addressing them one at a time. Realizing that Hila
was confused by the appearance of the
x(t) graph, Alon attempted to ease the
confusion by simplifying the input to a
one-dimensional motion, illustrating the
relationship between the planar motion
and the graph. Or perhaps he tried to create a constrained situation with the tool in
which the only possible second variable
in addition to x was time, allowing him to
argue the dependence between the two.
Alon was consistent in using the reductionist strategy. On the last demonstration,
he relaxed two variables, both the x and
y positions, allowing time to be the only
active component.
Further Answers
Hila still had her doubts and looked confused. She raised more questions about
the meaning of dependence and the notion of an independent variable.
16. Hila: In a coordinate system, time
is … time is a name of an axis.
17. Alon: Right.
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
18. Hila: You should have an
x-axis and a y-axis. If your
x-axis is the time axis, the
time just passes by itself.
The question Hila raised here seemed to
take a different direction from the problem
of dependence discussed in the previous
segment; in this instance it had to do with
symbols and mathematical representation
on the Cartesian graph. She could not see
the reason for time’s being represented
by an axis in the graph if they all accepted
that it did not depend on anybody. Alon
agreed that time was merely a name of
an axis (17), but then Hila argued that the
axes should be x and y. Hila continued
with a conditional statement (18) that
seems to indicate that she was again
bothered by the presence of uncontrollable time as an axis.
In another effort to help Hila distinguish
among the x-axis, the quantity that x
represented, and the time axis, Alon tried
a different strategy to expand his explanation of the meaning of the x-coordinate.
Alon’s Explanation of the x-Coordinate
19. Alon: What is an x-coordinate? You see x? X tells us
about width and length.
20. Alon: Have a look. When I move
to the right, it goes up, because
we are talking here about the
x-coordinates. And this is the
length. And if I move here …
21. Maya: Yes, he is right.
22. Alon: And if I move here
to the negative part, is the
x here the positive x?
23. Alon: Here it is the negative part of x.
24. Hila: I see.
25. Alon: If I move here, it will be
negative. Now I move to the
right, and it will be positive.
In this discussion Alon provided a geometrical metaphor: He looked at the x-y
plane as a rectangle with width and length,
choosing x to represent the length. Simultaneously analyzing the hand motion and
changes in the x(t) window, he argued that
the length he controlled with the mouse
was what they saw in the x(t) graph (20).
As in the previous segment, Alon tried
to simplify the explanation by removing
distractions, in this instance, the referencing of the axes, and he renamed x and y
as length and width. He also used new
terms to describe his motion: right and left
instead of forward and backward, which
he had used before (14). And he confirmed
that everyone understood the conventions
he was using (20). Maya, who was quiet
for most of the discussion, seemed satisfied (21). But Hila continued her questions.
Alon Explains
26. Hila: Then why do we
have time here?
27. Alon: Oooh.
28. Alon: Time is always there.
If there were no time, it
would have been a point. It
would not have moved.
Although Hila accepted the explanation
(24) about the connection between the
right and left sides of x in the x-y plane,
and between the positive and negative x in
the x(t) window, she did not connect it with
the role of time. As Maya and Aric were
laughing, Alon seemed exhausted and
ready to give up on convincing Hila (27).
Alon then suggested one final proof of the
presence of time: while he was holding
the mouse in one place, the x(t) graph
continued to move. If there were no time,
the graph would show a point.
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
The Emergence
of a Tool-Situated
Understanding of Time
The two episodes illustrate the use of
the tool to model a phenomenon that is
external to it (i.e., Hava’s ball) and to create a phenomenon that resides within the
tool (i.e., Alon demonstrating the duration
of time). The ways in which students use
the symbols within the context of the tool
show how they construct meaning. Do
mathematical symbols built in the tool
become part of the analysis of the given
physical phenomenon? And if so, how?
A central part of the analysis was learning how the students dealt with descriptions of the phenomena by the x(t) and
y(t) graphs and how, on the basis of these
graphs, they treated the presence of time
as a variable. Another important aspect of
the analysis dealt with the development of
the meaning of the graphs as part of coordinating temporal and spatial representations of time. Conversations within the
group and the intermingled actions with
the tool helped in tracing the students’
different grasps of the relationships among
space, time, and causality.
Describing Motion as Spatial Events
When reproducing the ball’s motion in the
first episode, most students viewed the
x- and y-axes as the floor and Hava. The
orthogonal setting of the graph system
naturally resembles the arrangement of
objects in reality, and the students seem
to have been acting instinctively, unaware
of important properties of the real phenomenon. Decisions on how to move with
the mouse seemed affected only by the
mental image of the shape of the ball’s
path and not at all by physical characteristics of motion, such as velocity, gravitation, or acceleration. Trajectories were
not drawn accurately, and the students’
work gave no evidence that shape was
an issue with them when they moved
the mouse. They may have performed a
kind of “pseudo-empirical abstraction,”
as described by Herscovics (1996). But
although the drawing did not look like
a conscious mathematical action, the
symbols (i.e., the Cartesian axes) provided
terms for interpretation.
Pointing to the x-y and y(t) graphs,
students retrospectively described the
phenomena mainly by its four events
(marked by the final position of the ball in
each phase): touching the ground, touching the wall, back on the ground, and back
in Hava’s hands. Duration was ignored;
for example, Aric and Alon disregarded
the fact that their graph represented the
time the ball spent on the floor on the way
up differently from the time it spent on
the way back. And students seemed to
completely dissociate the spatial order of
events from the time required to complete
them. Analyzing the x(t) graph, the students were still concerned with motion as
a function of the final position of the ball [
]. They did not yet point out that time was
a component of the phenomenon, nor that
it appeared as a variable in the representations. However, indicating that the ball
“stayed on the wall for a while,” Alon began to attend to duration, which was the
first instance of a student’s coordinating
the spatial and temporal characteristics.
Coordinating Spatial and
Temporal Characteristics
Coordinating changes of position with
changes in time is, according to Piaget
(1974), an essential component in understanding the concept of time. The two
episodes provide convincing evidence that
recognition of this connection is essential
and complex even for a 13-year-old. In the
beginning of the first episode, despite the
fact that the y graph appeared synchronically with the motion of the mouse, Alon
and Aric paid no attention to it. When they
noticed it, they explained it without effort,
although whether they were aware of the
change in the quantities represented in
the graph or were merely comparing the
shapes of the graphs is not clear. They
interpreted the y(t) graph almost with the
same narrative they had used when drawing. Time was absent from the conversation because they were concerned with
the succession of events but not with their
duration. When the x(t) graph appeared
with its surprising shape, for the first time
Alon become concerned not only with
space but also with time. Although he
returned again to the spatial succession
of events on this occasion he mentioned
duration: “Somewhere here it stayed on
the wall”; “stayed here for a while.” Piaget
(1974) helps explain the situation: “The
unicity of time that is lacking at first, appears quite naturally at a certain level of
intellectual development” (p. 49).
Like Alon, Hila also reconstructed overall
motion arranged in sequence by purely
spatial characteristics. But when the x(t)
plane introduced a new point of difficulty,
which could have been an opportunity
for expanding the understanding of motion and its components, Hila continued
to ignore the temporal characteristics of
the motion. She resisted associating the
spatial and temporal orders and struggled
with the presence of time, especially as
an independent variable: “Why does x
depend on …? What depends on us?”
Did Alon or any of the other students think
about the picture of the ball’s path as its
mathematical description? Most likely they
did not perceive the path in the x-y plane
as defined by two independent variables
changing in time. In the x-y plane, x and
y were locators, independent quantities
playing equal roles. In the x(t) and y(t)
planes, the locators surprisingly appear as
functions of time and represent additional
information. Hila was uneasy with this
representation of motion, especially with
x having changed from being at first independent to being a quantity that depended
on time. The representation of this additional information offered an opportunity to
make students conscious of the difference
between spatial descriptions and coordinated spatial and temporal considerations.
Alon and Hila were both conscious of
the appearance of the new symbols of
motion. Their reaction, although different
(Alon tended to deal with the temporal
characteristics that Hila struggled against),
steered the conversation toward the coordination of spatial and temporal components of motion. Schwartz (1996) calls this
consciousness the ability to distinguish
between the image for the eyes (i.e., the
picture of the path) and the image for the
mind (i.e., the graphs in time). Similarly,
Nemirovsky & Tierney (2001) emphasize
the different roles of the graphs when they
distinguish between seeing a (seeing a
path in the x-y plane) and seeing as (considering the path together with the motion
that generated it). This distinction helps
trace the development of the meaning
of time. The image for the mind, the x(t)
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
graph, although only vaguely connected
with the trajectory of the ball, helped explain the need for coordinating the spatial
and temporal characteristics. Following
Herscovics (1996), the present study suggests that a first phase of the abstraction,
the “separation of the concept from the
procedure (p. 359),” occurs here, situated
in the activity with the tool.
Psychological Time
Hila’s claim that “time just passes by
itself” reflects her awareness of the presence of physical time, which does not
depend on our actions. Piaget (1974) also
believed that “to the average adult, time
is, in fact, an ‘a priori form of sensibility’”
(p. 49).
Claiming that “these are things that don’t
depend on us,” Hila apparently made an
effort to separate physical time from the
construction of the path of the ball, and
thereby not to “psychologize” time. Time
was not a part of her reality while she
was working on “Hava’s ball.” Drawing
the ball’s path, she was convinced that
she alone was responsible for the x-y
coordinates, and obviously she was right.
According to Nemirovsky & Tierney (2001),
she formed homogeneous spaces—points
that represent locations and whose only
property is their position in relation to a
common origin and a set of axes. Although she operated the clock icon when
she started to record the motion, the
attention she paid to the drawing probably muddled the presence of the motion
she performed with the mouse. Hila was
concentrating only on spatial succession,
ignoring attributes of motion, such as
causality or duration. She did not explicitly
refer to psychological time and made no
mention of “now,” “before,” or “after.” For
her, therefore, psychological time, which
must be a part of the meaning of time according to Piaget, had likely not emerged
at that juncture.
Alon, demonstrating in the x(t) graph the
flow of time by activating the mouse and
holding it steady, “psychologized” his
private time. The purpose of using the
x(t) graph was to draw Hila’s attention to
physical time and to represent concrete
duration by referring it to mere psychological time. But he was considering his
psychological time, and its external representation probably did not trigger Hila’s
psychological time.
Time as an Independent Variable:
The Realization of Duration
Hila did not seem to understand why time
suddenly appeared. Because she moved
the mouse as she wished, and time did
not affect the motion she created (i.e.,
time does not cause the mouse to go up
or down), Hila could not see how time
played any role in the motion. She was no
doubt aware of the existence of time in the
context of temporal processes, but in this
activity, she did not recognize any characteristics of time (such as duration or temporal order) that the graph x(t) symbolized.
For her, the crucial component of motion
was the spatial order in her drawing.
An additional interpretation of Hila’s inability to grasp time as a component in
the motion of the ball is based on the role
of time as an independent variable. Her
previous experience of sketching penciland-paper graphs from points contradicted the graphs constructed by the tool.
In previous graphing tasks (not necessarily mathematics tasks), the horizontal
axis had been assigned to independent
variables, such as weight, age, and
scores. Her role had been to control these
variables by choosing which data to plot,
often by assigning discrete values (e.g.,
0, 1, 2, 3) to the independent variable. For
her an essential quality of determining the
independent variable was choosing it freely when creating the graphic representation. Her notion of an independent variable
was “any quantity that I determined.” But
in the drawing and graph created with the
tool, no values were assigned, making her
role entirely passive regarding the scale
and values of the independent variable.
This discrepancy between the conception
of time as a free, uncontrolled variable
and the conception of the independent
variable as something she could designate
may have prompted her question: “Why
does x depend on …? What depends on
us?”
Alon, however, demonstrated the active
and independent role of time by holding the mouse motionless in one point
and holding down the mouse button
that activates the graph. He succeeded
in constructing the ideas of succession
and duration. Holding the mouse steady
while time passed, he surprisingly created
from one event a temporal succession of
events. He neutralized the spatial succession (that is, no events occurred in space)
and emphasized the role of time and that
of duration. Piaget (1974) claims that to
follow time along the simple and irreversible course of events is simply to use it
without cognizance of it. He also infers
that continuity of time should not be taken
for granted at all levels of mental development, rather, “the continuity of time calls
for a special construction” (p. 273). Alon
managed to produce a special construction that allowed a visual representation
of the pure flow of time, and he thereby
constructed a concrete representation
of an action that is usually accepted as
purely mental.
When Alon held the mouse to demonstrate
the time flow, he was abstracting, although
his action may not fit the definition of a
formal abstraction. His interaction with the
models demonstrated his appropriation of
the tool and its integration with the activity.
Alon challenged Hila’s image of time and
dependence by producing representations that contradicted her mental image.
To do so, he changed the built-in schema
of the tool to a different one, which was
suited for his purpose. Grasping the correct meaning of the graphs, he designed
an experiment in which he subordinated
the symbols of the tool to his interpretation and planned an event that produced
specific x(t) and y(t) graphs to support his
arguments. In this way, he expressed his
idea, proceeding from the mathematical
argument he wanted to make, through
an image of a phenomenon he wanted to
create, and to the mathematical presentation of that phenomenon. The tool did not
explicitly support his action, but rather,
the opposite. But beyond this mental
reversibility, which is often described as
a characteristic of abstraction (Herscovics, 1996), and beyond the redefinition of
the roles of participating representations
in the experiment, which is considered as
another aspect of abstraction (diSessa,
1987), yet another aspect of abstraction
is apparent—one that is unique to the
concept of time: Alon not only generated
another event with the tool but also, distilling a notion from the story, succeeded in
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
constructing a demonstration of a phenomenon that reflects the invariance of
space in the meaning of time. He thereby
completely abstracted the spatial characteristics and concentrated on the essential
property of time—its duration.
A Tool-Situated Emergence
of the Qualities of Time
occurred and the tool began to be used to
fulfill roles in planned mathematical reasoning about the mathematical symbols
of time.
This paper explores how students construct the concept of time when they first
encounter its mathematical representations. The focus is on the emergence of
a tool-situated understanding of time:
from colloquial and intuitive uses of time
to its mathematical representation in
graphs. Analyzing the episodes from a
tool perspective exposed epistemological and cognitive aspects of meanings of
time that emerged using the symbols built
into the tool. The emergence of meaning
of mathematical symbols of time observed
here seems similar to the emergence of
the perception of time described by Piaget
(1974). Thus the cognitive obstacles to the
creation of meanings of the mathematical
symbols of time appear to have their origin
in epistemological complexities of the
concept of time. Beyond the opportunities
that the technology opened for students
to construct meaning for representations
of temporal processes in unique ways, the
technology provided an opportunity for us
to learn new aspects of the complexities
involved in symbolizing time in Cartesian
graphs.
The capability of technology to support
students in experimenting with temporal
processes by means of naive actions and
in creating these processes to produce
proper mathematical symbols of space
and motion played a major role in this
study.
From the epistemological perspective,
we conclude that the properties of time
cannot be easily perceived or determined.
But before one can begin to think about
time in mathematical terms, a complexity that is unique to time requires that one
coordinate the different meanings of time
(physical and psychological time, time
as duration, and so on) in the different
situations in which time is observed. An
understanding of time emerges from the
ability to coordinate these different meanings. In the cognitive dimension, this study
sheds light on the process of tool-situated abstraction that reasoning about the
mathematical model of time undergoes
with the aid of a tool. These tool-situated
abstractions emerged from the attempt to
assign meaning to the correlation between the different symbols (i.e., graphical
representations) of the same phenomenon
(i.e., “Hava’s ball” or Alon’s impromptu
drawings). The abstraction evolved from
an initial interpretation of the mathematical symbols of the tool using the everyday
terms of the phenomenon, until a change
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
Appendix A: Transcript
Illustration of the Software
Drawing the Picture
1. Alon: In effect, we are drawing
“Hava,” what happens to her ball.
2. Aric: Now we go up.
Here is Hava.
3. Alon: It touches the floor, then
hits the wall, goes back down,
and comes back to Hava.
Drawing y(t)
4. Alon: Well, they say here, “Hava
likes to play with the ball. She
throws it to the ground. The ball
hits the ground and then it hits
the wall.” So, here it hits the
wall, goes back to the ground,
and then goes back to Hava.
5. Alon: Here is Hava. The
graph says it goes down.
6. Alon: It goes up, it hits the
wall. It goes down. Because
of the wall, it falls down and
goes back to Hava’s hands.
Drawing x(t)
7. Alon: The x-coordinate.
It is not important.
8. Alon: Oh, that’s hard to
understand. Hmm.
9. Alon: I think that here it is Hava’s
ball. She threw it to the ground.
Here somewhere it stayed
on the wall. It hit the wall and
stayed here for a while, just
stayed in the middle simply
because of “time,” and then
it bounced back to Hava.
10. Alon: And here it is just an x.
Hila’s Question
Alon’s Explanation
11. Hila: Why does x depend
on …? What is dependent
on us? Or independent?
26. Hila: Then why do we
have time here?
12. Hila: For instance, time. It
passes. We can’t stop it. Or,
for example, our growth. We
can’t stop it. These are things
that don’t depend on us.
28. Alon: Time is always there.
If there were no time, it
would have been a point. It
would not have moved.
Alon’s Answers
13. Alon: It is not exactly …
14. Alon: I’m taking it back, and then
it goes down to the negative part.
I’m taking it forward, and then
it goes up to the positive part.
15. Alon: And if I don’t move it,
“time” will continue to pass.
Time’s Role on the Axis
16. Hila: In a coordinate system, time
is … time is a name of an axis.
17. Alon: Right.
18. Hila: You should have an
x-axis and a y-axis. If your
x-axis is the time axis, the
time just passes by itself.
Alon’s Explanation of the x-Coordinate
19. Alon: What is an x-coordinate? You see x? X tells us
about width and length.
20. Alon: Have a look. When I move
to the right, it goes up, because
we are talking here about the
x-coordinates. And this is the
length. And if I move here …
21. Maya: Yes, he is right.
22. Alon: And if I move here
to the negative part, is the
x here the positive x?
23. Alon: Here it is the negative part of x.
24. Hila: I see.
25. Alon: If I move here, it will be
negative. Now I move to the
right, and it will be positive.
27. Alon: Ooh.
Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
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Journal for Research in Mathematics Education
Medium and Meaning: Video Papers in Mathematics Education Research
Credits
Authoring
Authors:
Michal Yerushalmy, University of Haifa
Beba Shternberg, Center for Educational Technology (CET)
Based on:
Yerushalmy, M., & Shternberg, B. (March 2000). Bridging modeling to algebra, v 1.0 [Videopaper]. Presented at
Videopapers in Mathematics Education Conference 2000. Dedham, MA.
How to cite this work:
Yerushalmy, M., & Shternberg, B. (2001). Epistemological and cognitive aspects of time: A tool perspective. v3.0
[Videopaper]. In Medium and meaning: Video papers in mathematics education research, Journal for Research in
Mathematics Education Monograph, Vol. XIII. 2005. [On CD-ROM]. 5 min. video (24.3 MB), 25 pages text (95 KB),
19 images (415 KB).
Acknowledgements:
The authors are grateful for comments on earlier drafts by Rina Hershkovitz and by anonymous reviewers.
Technical production
Original footage by:
Beba Shternberg
Original footage format:
VHS
Original footage date:
November, 1998
Digital movie editing:
Digitizing, editing, and compressing for version 1.0, Shmuel Klimovsky
Digitizing, editing, and compressing for version 2.0, Darrell Earnest
Digital movie settings:
All digital clips used in this videopaper are QuickTime 4.0 movies.
Compression settings: Sorensen CODEC, 24 fps, key frames every 15 frames, 560 kbs/s
Encoding software: Media Cleaner 5.0
Images:
Shmuel Klimovsky, Beba Shternberg, Michal Yerushalmy
HTML:
For version 1.0, Shmuel Klimovsky
For version 2.0, Darrell Earnest
For version 3.0, Cara DiMattia
Transcription and
Translation
Talma Mokedi, Beba Shternberg
Closed captioning:
For version 1.0, Studio Videospot, Israel
For version 2.0, Darrell Earnest
For version 3.0, Cara DiMattia
Videopaper design and
assembly:
Design for version 1.0, Shmuel Klimovsky, Beba Shternberg, Michal Yerushalmy
Design for version 2.0, Darrell Earnest, Beba Shternberg, Michal Yerushalmy, using VideoPaper Builder
v.99
Design for version 3.0, Cara DiMattia
Function Sketcher
Software Design:
Beba Shternberg, Michal Yerushalmy
Function Sketcher
Software Programming:
Mike Medved, Gil Semo, Alexander Zilber
Production Software:
Media100, VideoPaper Builder v 0.99, Photoshop 6, Dreamweaver MX, QuickTime 5
Grant information
Support for the multimedia version of this paper was provided by NSF Grant 9805289, “Bridging Research and Practice,” awarded to Ricardo Nemirovsky and David Carraher. All opinions and analysis
expressed herein are those of the authors and do not necessarily represent the position or policies of the
funding agencies.