ORIT ZASLAVSKY, HAGIT SELA and URI LERON
BEING SLOPPY ABOUT SLOPE: THE EFFECT OF CHANGING THE
SCALE
ABSTRACT. What is the slope of a (linear) function? Due to the ubiquitous use of
mathematical software, this seemingly simple question is shown to lead to some subtle
issues that are not usually addressed in the school curriculum. In particular, we present
evidence that there exists much confusion regarding the connection between the algebraic
and geometric aspects of slope, scale and angle. The confusion arises when some common
but undeclared default assumptions, concerning the isomorphism between the algebraic
and geometric systems, are undermined. The participants in the study were 11th -grade students, prospective and in-service secondary mathematics teachers, mathematics educators
and mathematicians – a total of 124 people. All participants responded to a simple but nonstandard task, concerning the behavior of slope under a non-homogeneous change of scale.
Analysis of the responses reveals two main approaches, which we have termed ‘analytic’
and ‘visual’, as well as some combinations of the two.
KEY WORDS: Cartesian coordinate system, cognitive conflict, computers, function, graph,
graphic software, representation, scale, slope
1. I NTRODUCTION
Our study was triggered by the increasing use of graphic software for
learning about functions, where students are often invited to freely zoom in
or out, since supposedly this “doesn’t change the behavior of the functions
under investigation”. They look at very different figures and are expected
to see a common behavior ‘through’ them.
But a number of studies point to some illusions and pitfalls that students may encounter when working with graphic software (Demana and
Waits, 1988; Goldenberg, 1988; Hillel et al., 1992). These technologies
often reveal conceptual obstacles associated with the connection between
analytic and geometrical representations of functions. Yerushalmy (1991)
addresses the importance of representing the same function in various homogeneous and non-homogeneous coordinate systems and identifying all
graphs as representing the same function. Clearly, being able to perceive
invariance through the specific visual features of each representation is an
indication of understanding and a step towards abstraction. However, this
approach seems to overlook the implications of treating the angle between
Educational Studies in Mathematics 49: 119–140, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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ORIT ZASLAVSKY ET AL.
the graph and the horizontal axis in various systems as ‘non-relevant visual
information’ (ibid, p. 49). The present study highlights the significant (albeit problematic) role played by the angle between the horizontal axis and
the graph of a linear function and its relation with the notion of slope.
Problems of understanding the notion of slope are addressed in a number of studies, mostly in connection with the common misconception of
confusing height for slope (Leinhardt, Zaslavsky and Stein, 1990). Some
studies, like the present one, treat slope as a purely mathematical entity,
while others embed it in a physical or ‘real life’ context (Nemirovsky,
1992, 1997; Clement, 1989). Rasslan and Vinner (1995) investigated
“whether students realize that the slope is an algebraic invariant of the line
and therefore does not depend on the coordinate system in which the line is
drawn” (p. 264). This formulation implies no clear distinction between the
slope of a function and the slope of a line. Our study demonstrates the need
for making this distinction. Schoenfeld, Smith and Arcavi (1993) describe
in detail the subtleties involved in one student’s attempts to make sense
of the slope of a linear function through interaction with mathematical
software. Their study points to the depth and complexity of this concept
in its traditional view. Our study adds yet more complexity and subtlety to
this picture.
In this paper, we examine some implicit assumptions concerning the
connections between the algebraic and geometric systems within which
we operate in representing linear functions and analyze the notion of slope
from mathematical and pedagogical perspectives.
2. T HE STUDY
2.1. The task
In our work with students and student teachers, we have developed the following two tasks in order to investigate the mathematical and pedagogical
implications of scale change in coordinate systems.1
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BEING SLOPPY ABOUT SLOPE
Task 1:
The graph in Figure 1 represents
the function f such that f:x→x
Task 2:
A student plotted the graph of the
same function f (such that f:x→x)
in a computerized environment
and got the following graph (Figure 2).
f(x)
f(x)
9
3
6
2
1
-3
-2
-1
3
"
1
2
3
x
-1
-3
-2
-1
-6
-3
-9
Please respond
1–3 below.
to
2
3
x
-3
-2
Figure 1
$
1
Figure 2
questions
Please respond to questions 1–3
(the same as in Task 1) with respect to Figure 2. Describe your
considerations, reactions, dilemmas and other thoughts regarding
these questions.
For each of the above Tasks answer the following questions:
1. What is the slope of the function f ? How did you determine it?
2. Does the graph of f bisect the angle between the axes? How do you
know?
3. Can you find the tangent of the angle between the graph and the
x-axis? If you can – what is its value? How did you calculate it?
If not – why not?
Task 1 served as a baseline for comparison, while Task 2 was designed
specifically to trigger re-examination of the notion of slope (e.g., is it 1 or
1/3?).
We first establish some conventions and terminology. All coordinate
systems in this paper are assumed to be orthogonal. We distinguish between
a homogeneous coordinate system, where the units on both axes are the
same and a non-homogeneous coordinate system, where the units are diffe-
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ORIT ZASLAVSKY ET AL.
rent.2 For example, if the unit on both axes is 2 the system is homogeneous;
if the unit on the x-axis is 2 and on the y-axis 3, the system is nonhomogeneous. Note that the system appearing in Task 1 is homogeneous,
while the one in Task 2 is non-homogeneous.
2.2. The subjects
There were 124 participants in the study and they were distributed among
5 different groups: 28 eleventh-grade students studying calculus; 28 preservice secondary mathematics student-teachers; 50 experienced in-service
mathematics teachers; 14 mathematics educators (that is, graduate students involved in thesis research and in teacher education); and 4 research
mathematicians. All participants were given a written questionnaire consisting of the above two tasks. In addition, 12 participants were interviewed
as they responded to the tasks and others took part in group-discussions
following these tasks.
The analysis of the data – written responses, the transcribed interviews
and field notes of the group-discussions – focused on the following aspects: Implicit and explicit concerns of the participants, their assumptions
and approaches and the nature of the experience some of them have gone
through while trying to cope with the attendant cognitive conflict.
3. A NALYSIS OF THE SUBJECTS ’ SOLUTIONS OF THE TASKS
Although the two tasks appear to be quite simple, they have proved to be
quite unsettling for the participants and generated considerable excitement,
confusion, uneasiness and disagreement.
The answers to Task 1 (as expected) were the same for all participants,
thus requiring no special analysis; as mentioned above, Task 1 mainly
served to set the ground for the subtleties of Task 2. The responses to Task
2, in contrast, varied quite dramatically. In analyzing the various responses
to Task 2, we have found two main perspectives of slope, which we have
termed ‘analytic’ and ‘visual’, as well as certain combinations of the two.
We now describe these perspectives in more detail.
3.1. Analytic and visual perspectives of slope
In analyzing the responses to Task 2, two main perspectives on slope were
identified, which we have called analytic and visual (cf. Table 1). According to the analytic perspective, slope is a property of the function and
does not depend on the function’s representation; in particular, the slope
should be invariant under non-homogeneous change of scale. The slope of
a linear function, in this view, can be computed either as derivative, or a
BEING SLOPPY ABOUT SLOPE
123
difference-quotient, or the coefficient of x in the equation y = mx+b. The
visual perspective, in contrast, takes the slope to be a property of a line
(the graph of the function), which may thus vary under a change of scale.
The slope of a linear function, in this view, is determined by the angle
α between the x-axis and the line representing the function; that is, the
slope is tan(α), or the quotient ‘rise over run’. We note that the distinction between the analytic and visual perspectives is closely related to the
distinction between representational and iconical (or literal) interpretation
of graphical information in Pimm (1995). This alternative analysis will be
discussed further in Section 3.4.
TABLE I
Analytic vs. visual perspectives of slope
Analytic perspective
Visual Perspective
Slope is a property of the (linear) function.
Slope is an invariant under non-homogeneous change of scale.
Slope is a property of the graph of a
(linear) function.
Slope varies under non-homogenous
change of scale: different lines representing the same linear function may
have different slopes.
Slope is computed as:
tan(α);
quotient of segment-lengths.
Slope is computed as:
derivative;
difference-quotient;
coefficient.
3.2. Slope in a non-homogeneous system: Different approaches to Task 2
Our data shows at least two different types of answers for each question
in Task 2 (This is true even within each of the 5 groups of participants).
Table II presents the total distribution of the answers to Task 2 (with the
exception of one participant, whose answers did not fall into any of the
categories).
As seen in Table II, two different answers were found for Question
1, two for Question 2 and three for Question 3. In analyzing the written
responses to Task 2, the three answers given by each individual to the
three questions were taken as a single entity, reflecting this individual’s
approach to the problem. Of the 12 formally possible approaches (triplets
of answers), only four were actually taken by the participants. In addition,
one participant questioned the legitimacy of the questions in Task 2 (see
section 3.3). A close look at the written explanations and an analysis of the
124
ORIT ZASLAVSKY ET AL.
TABLE II
Four main approaches taken in response to Task 2 (the text boxes below the Table give the
underlying characteristics of each approach)
follow-up interviews enabled us to characterize each approach together
with its underlying assumptions and beliefs.
As seen in Table 2, approach A reflects an Analytic perspective, while
approach V reflects a visual perspective. Approaches M1 and M2 are different ‘mixes’ of the analytic and the visual perspectives. Note that different
approaches can yield the same answer to a particular question.
Approach M1 differs from the other approaches in the uncertainty expressed regarding the answer to question 3: Can tan(β) be calculated at
all? And if it could, should its value be 1 or 1/3?
BEING SLOPPY ABOUT SLOPE
125
Approaches A and V share the view that in any coordinate system the
slope of a linear function is equal to the tangent of the angle between its
graph and the horizontal axis (α in Figure 1, β in Figure 2). Approach A,
however, reflecting as it does a ‘pure’ analytic perspective, differs from
the other approaches by considering even an angle to be a purely analytic entity, while the other approaches treat the angle between lines in a
coordinate system as a geometric entity.
We now discuss in more detail each of the approaches and their underlying concepts and beliefs, together with some illustrative excerpts from
the interviews.
Approach A
As mentioned above, this approach reflects an analytic perspective. The
1
. It is
slope of a linear function is perceived as the difference-quotient yx22 −y
−x1
the quotient of ‘pure’ numbers and does not depend on any representation
of the function. It is based on the assumption that in any coordinate system
this difference-quotient is equal to tan(α), α being the angle between the
graph of the function and the x-axis. However, tan(α) is considered a purely
analytic entity which is, in effect, defined by the above difference-quotient.
Bella, a doctoral student in mathematics education with a strong mathematical background, took this approach:
Interviewer:
Bella:
Interviewer:
Bella:
Interviewer:
Bella:
Does the line bisect the angle between the axes?
Yes. β=45◦.
Is this an angle of 45◦ ? [points to angle β in Figure 2]
Yes. This is a representation of 45◦ in this coordinate system.
Does this mean that this angle [points to the angle between the
line and the y-axis] is also 45◦ ?
Yes. The length of the segment AB [marks points A, B, and O, as
in Figure 3] which looks like 1 is really 3. In triangle OAB you
can calculate the tangent of β as follows:
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ORIT ZASLAVSKY ET AL.
T an(β) =
AB
3
= = 1.
OA
3
Figure 3.
Interviewer:
Bella:
Is the angle in Task 1 congruent to the angle in Task 2?
Yes. You are using a visual approach. There are cases in which
not everything that looks like a certain mathematical object is
what it appears to be. For example, if you represent on paper
a circle in a three-dimension structure, it looks like an ellipse
[draws Figure 4]:
Figure 4.
Or another example: When we sketch a box on paper, we treat
certain angles as being 90◦, although in the figure they look like
acute or obtuse angles [draws Figure 5]:
BEING SLOPPY ABOUT SLOPE
127
Figure 5.
Interviewer:
Bella:
If you are in analytic geometry you must measure angles analytically.
How do we measure angles analytically?
For example, you can draw vectors [draws a vector on the line in
Figure 2 and marks (4,4) and (1,0), as in Figure 6]:
Figure 6.
There is a definition, a theoretical definition of an angle between
vectors that ignores the picture, and is:
4·1+4·0
cosβ = √
√
2
4 + 42 · 12 + 02
This defines the angle. And if you ask the question here, you get
the angle. You get in both cases 45◦ .
Bella, like others in the A category, did not seem to experience any special
conflict. She was able to make a clear separation between her algebraic and
geometric worlds and had no trouble treating an angle as an analytic entity
in the context of Task 2. It did not seem to contradict her geometric notion
of angle. She held a coherent view and was able to justify her responses.
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Approach V
Approach V, the least frequent among our participants, represents a purely
geometric perspective (see Tables I & II). In this view, the slope is defined
as tan(β), thus varies under change of scale. Benny, one of the mathematicians, has responded to Task 2 consistently, but felt uneasy about the
slope not being invariant under change of scale.
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
Interviewer:
Benny:
The thing that is confusing or thought provoking in this question
is that you can measure this in two systems of units: One can talk
of the y’s and the x’s and then it [the slope] really remains 1. This
means that an increase in a x-unit increases the y by one unit. And
here an increase in 3 cm in x increases by 1 cm the y.(. . .) This is
a matter of definition. If we just ask what the slope of the graph
of f: x → x is, and say nothing else, the understanding is that we
are in equal units.
And if not, is it not uniquely determined?
Maybe not. I am just thinking of this for the first time [laughs].
(. . .)
Someone said that there was no point in talking about tangent
because there is no sense in talking about angles [in Task 2]. What
would you say about this?
I don’t think it is true. The line exists and the axes exist and we
ask what the angle between two lines . . . This question is well
defined: There is a line. Here it will not be right to think [. . .] that
we are talking about [slope equals] 1. Because the line [x-axis]
and the graph are given.
But when we talk about slope, are we . . . talking about how it
looks?
I think so, because it is a line. The line is given. We have 3 straight
lines [the two axes and the graph]. But what is more important is
that this line [the x-axis] is given and this line [the graph] is given
and we ask what the angle is.
So you feel that slope is a property of the graph?
A geometric property.
Not of the function?
Right.
And the slope is not necessarily the derivative?
[Only] in a [homogeneous] Cartesian system it is the derivative.
But in this system it isn’t?
Right.
Pimm (1987) discusses the influence of spoken language on the use and
interpretation of mathematical terms. The visual approach to slope can be
BEING SLOPPY ABOUT SLOPE
129
regarded as influenced by the common usage of the word slope in everyday
context, where we usually regard the slope as the (tangent of the) angle
made between a straight line and the horizontal line.
Approach M1
This approach, the most common one among the participants, was marked
by a strong sense of disequilibrium and cognitive conflict. Stemming basically from an analytic perspective, it shares with approach A the view that
in any coordinate system the slope is equal to tan(α) (α being the angle
between the graph of the function and the x-axis) but, contrary to approach
A, takes an angle to be a geometric entity, as in approach V. These assumptions inevitably lead their proponents to a contradiction: Considering the
slope of the function in Task 2 as the difference-quotient implies that its
value must be 1. Assuming that the slope is equal to tan(β), implies that
tan(β)=1, hence β=45◦ . However, treating the angles in Figure 2 geometrically, clearly yields that the graph does not bisect the angle between the
axes, thus β cannot measure 45◦ .
Some people in the M1 category were not aware of this contradiction
and at first did not sense any cognitive conflict, but were later prompted
to re-consider their reply either through peer discussions or by the interviewer. Consequently, at the end, most of them abandoned their original
view. Those who noticed these contradictions from the start were confused
and could resolve the conflict only after discussions with peers or with the
researchers; and even after those discussions many remained puzzled and
uncomfortable. Those who eventually managed to resolve their conflict did
it by adopting either approach M2 (the vast majority) or A (only a few), or
by declaring as illegitimate questions about angles and their tangent in a
non-homogeneous coordinate system.
The following excerpts from the interview with Iris, an experienced
math educator and a doctoral student, demonstrate the conflict and struggle
typical to the approach M1 :
Interviewer:
Iris:
Interviewer:
[reading from the written answers to Task 2] [You wrote that] the
slope is 1, the line does not bisect the angle. How do we calculate
the slope? How can we be sure that it is 1?
Because I did the quotient of the y-values to the x-values. The
y
. Is that right?
[slope] m is x
I can ask what the slope of the function is only if I define that,
eh. . ., I take the same scale for both axes.
This means the slope is a property of the graph, not the function?
130
Iris:
Interviewer:
Iris:
Interviewer:
Iris:
Interviewer:
Iris:
Interviewer:
Iris:
ORIT ZASLAVSKY ET AL.
I don’t know what to say. I have to think. . .. We define a slope as
the derivative. . . . this means that the slope remains 1.
[points to the correspondence rule f : x → x and writes 1 as the
coefficient of x].
I could call this 1 [points to the 3 on the x-axis] and then that
[points to the 3 on the y-axis] would be 1/3. This means, that the
leg will be 1/3 of the length of the. . . the. . . horizontal leg.
Actually, what I said is that, if the quotient is 1, then the ratio
should be 1 to 1, but with respect to the centimetric size it is not
1. One is three times larger than the other.
OK. What about the bisecting of the angle? How can we be sure
whether it [the line] bisects the angle? You said it does not bisect
the angle.
One moment. Actually [if] it bisects I should have gotten 1
[laughs in embarrassment]. Now I am really confused. . . . Just
tell me that I’m not the first [to be confused]. [laughs]. . . This
presents a conflict, doesn’t it?
y
. Now, I,
One moment. One thing is for sure. The slope is x
eh. . ., am supposed to measure in the same units the difference in
y’s to the difference in the x’s.
. . .That is, for every unit on one leg I must have the same unit
on the other leg. Actually, what is happening to me here is as if I
have 1 to 3. If I go according to this scale and the ratio is 1:3. . .
It does not give me tan(45◦).
And this ratio gives you the slope?
[Thinks for a while. Seems to have difficulty to answer] I don’t
know what to say [laughs]. I need to think. I am embarrassed.
What is sure is that the slope is 1. The derivative is 1. No doubt
about it. . ..
Does it [the line] bisect the angle?
No, here it doesn’t bisect the angle.
OK, and how about the tangent. Can you calculate it?
The tangent is, we said, 1 /3 .
... ... ... ...
I am trying to think how to explain this conflict between the
derivative that equals 1 and the tangent that now I did not get
1.
Iris, like many others who have exhibited approach M1 , experienced a
considerable conflict. Her way of resolving the conflict was to abandon
her preliminary assumption that the slope is always equal to tan(β), while
still keeping an analytic perspective of slope and the notion that angles in
a coordinate system are geometric entities. In fact, at the end of the inter-
BEING SLOPPY ABOUT SLOPE
131
view she converted to approach M2 (see below), still feeling considerable
uneasiness.
Another way in which a number of people resolved their conflict was
by claiming that in a non-homogeneous coordinate system either α or
tan(α) were not defined. Joseph, another mathematician, demonstrated this
attitude:
Joseph:
Interviewer:
Joseph:
Interviewer:
Joseph:
Interviewer:
Joseph:
Interviewer:
Joseph:
[reads the 2nd question in Task 2] This is a famous fight, ah?
Now comes the famous fight. . . Can you calculate the tangent?
[reads to himself the 3rd question]. This is what the fight is about!
[laughs]. It’s a senseless question.
Why?
Because if. . .OK, it depends on the context. Let’s be more precise. If we are at the context of analytic geometry and you look
at the axes, there is no sense talking about the angle, because the
units are not equal. If you go out from this and look at it only as
a picture, as a geometric picture, so you can calculate the tangent
and it is 1/3. But it is senseless to ask it in this context.
But you answered before, that the line doesn’t bisect the angle,
so there is an angle here. You see it, don’t you?
Of course, I admit, I fell in this trap [laughs]. You can say here [Q.
2] that in analytic geometry this is a senseless question, because
you don’t speak about geometry, but about algebra. You can start
here [Q.1] – What is the slope? The slope is 1. What is the angle
here? It depends, if you are in analytic geometry, so you have to
measure angles analytically.
But we see here an angle, so what is its measure?
If you look at it geometrically, the angle is 33◦ , or whatever. . . If
you look at it as an analytic problem, the angle is 45◦. That’s the
reason I say it’s a senseless question. Because it is 45◦ , but I see
33◦ , or whatever. . .
How can the same angle be of different measures?
You just don’t talk about it. [laughs]
It’s a senseless question. It’s like talking about apples and pears.
It’s like dividing by zero. You don’t do it.
Approach M2
Approach M2 differs from M1 in its perception of the relation between the
slope of the function and the angle α (spanned between the x-axis and the
graph of the function). While according to M1 the slope is always equal
to tan(α), according to M2 this is true only in a homogeneous coordinate
system.
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ORIT ZASLAVSKY ET AL.
Those who exhibited this approach did not experience any conflict in
answering the three questions of Task 2. Their answers were consistent
with the view that emphasizes invariance under change of coordinate system. The answers were also consistent with the view that angles should
always be treated as geometric entities.
Saul, a mathematician, exhibited this approach. Although he had no
doubts about the answers to Task 2, he confessed that the task was challenging. He had to stop and think about certain assumptions that were
taken for granted. Saul insisted that the slope is a property of the function
and cannot depend on the coordinate system in which it is represented.
Moreover, he felt certain that mathematicians should agree on a definition
of slope and the answers to Task 2. When we told him that his colleague
Benny has chosen a visual approach (see above), he was determined to
convince Benny that he was in error.
We turn to Saul’s attempts to subsequently convince Benny to change
his mind.
Benny, who at first manifested Approach V (see above), was willing to
‘convert’ to M2 after a long discussion with Saul over the phone. Here is
an excerpt:
Saul:
[Calls up Benny] The person who talked to you about slope is
sitting here with me now. I’ll tell you what I told her. I said that the
slope of a line is taking two points on it, taking the differences of
the y coordinates to the differences of x and this does not depend
on the coordinate system. It can’t always be the tangent of the
angle. It is the tangent of the angle only when the axes are the
same. I proposed to define the slope as the difference quotient
and not the tangent of the angle.
[Benny says something on the phone and Saul replies]. No, there
is no problem, because you take a point. The point is the same
point in the system. You take 1 and it remains 1. If you take (1,1)
in a regular system [homogeneous system in our terminology], it
looks in the picture in the middle and it looks not in the middle in
a system where you changed the scale. But it is still (1,1). When
you take coordinates it doesn’t matter (. . .) the slope should not
depend on the system. The tangent does depend
(. . .)
[Replies to Benny]. She [the interviewer] confused you, she confused me . . . We can’t give her a confused answer, right? (. . .)
Look, the point is, that in my opinion the slope should not depend on (. . .) So the difference is exactly . . . there is a difference
between slope and tangent. Tangent is a geometric notion.
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BEING SLOPPY ABOUT SLOPE
The last excerpt points to the main difference between the visual and the
analytic approaches: the question of invariance of slope under change of
scale. Benny was in the end convinced that there should be an agreement between mathematicians on the definition of slope and was willing moreover to base this definition on the principle that slope should be
invariant under change of scale.
3.3. Distribution of approaches
Altogether we have collected written responses from 5 groups of people,
totaling 124 participants. Table III is a break up of the total in Table II into
subgroups. It displays the distribution of the different approaches to Task
2 among the groups.
From Table III we can see that for almost all the groups, the visual
approach V was the least popular while the mixed approach M1 the most
popular. One mathematics educator, Nora, responded in a way that did not
fit any of the categories, thus we did not include her response in Tables II
and III.
Nora’s way of resolving the conflict was by restricting the kind of system in which it is legitimate to represent functions. Perhaps she was groping at the idea that non-homogeneous representations should be handled
with care, since angles and distances do not remain invariant in them.
TABLE III
Distribution of approaches to Task 2 by groups
Participants
Approach Analytic
A
Mixed
M1
M2
Visual
V
Total
High school students
9 (32%) 9 (32%)
7 (25%)
3 (11%) 28 (100%)
Student teachers
9 (32%) 14 (50%)
5 (18%)
0 (0%)
28 (100%)
Inservice math teachers 2 (4%) 22 (44%)
22 (44%)
4 (8%)
50 (100%)
Mathematics educators
3 (23%) 5 (38.5%) 5 (38.5%) 0 (0%)
13 (100%)
Mathematicians
1 (25%) 1 (25%)
1 (25%)
1 (25%)
4 (100%)
Total
24 (20%) 51 (41%)
40 (33%)
8 (6%) 123 (100%)
3.4. Looking through another lens: Representational versus iconical
interpretation3
Approach A is clearly a manifestation of treating visual information, namely,
the graph of a function, as a representation of an algebraic entity. On the
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ORIT ZASLAVSKY ET AL.
other hand, Approach V may be interpreted as treating the visual information iconically (or literally), ignoring what it stands for (Kerslake, 1981;
Clement, 1989; Leinhardt et al., 1990; Pimm, 1995). On this interpretation,
the mixed approaches may reflect a shift back and forth, due to uncertainty,
from looking at the graph representationally vs. iconically.
However, given the special background of the participants in our study,
we are reluctant to suggest that they were looking at the graphs iconically.
The majority of the participants (78%) had taken at least some undergraduate mathematics courses. Over 50% of the participants had at least a B.Sc.
in mathematics. Among the participants who held approaches V , M1 or
M2 , there were mathematics educators and mathematicians with a very
strong mathematical background. Thus, we tend to think that they were
not looking at the line in the coordinate system merely iconically. Rather,
they were probably following what is a standard procedure in geometry:
relying on the appearances of the graph to a certain extent, but taking into
account that it only represents certain features of the function. Like any
standard mathematical procedure, such performance eventually becomes
automated and, in fact, serves us quite well under normal circumstances.
The point of our tasks, however, was precisely that it did undermine some
default assumptions, which are part of those ‘normal circumstances’, hence
the confusion that was the share of even these experienced mathematical
practitioners.
4. A NALYSIS OF THE TASKS FROM A MATHEMATICAL POINT OF VIEW
In this section we offer our own analysis of the mathematical issues raised
by Task 2.4
4.1. The dilemma
The dilemma concerns the notion of slope and what happens to it under a
non-homogeneous change of scale. On the one hand, if we assume, as we
normally do, that slope is a non-ambiguous property of a (linear) function,
then it should be invariant under change of scale: All lines representing
the same function should have the same slope. On the other hand, one can
clearly see (cf. Tasks 1 and 2) that the slope actually changes. So how can
we reconcile the two conflicting views? Should the slope in Task 2 be 1 or
should it be 1/3?
4.2. Mathematical background
In our view, the fundamental notion behind the whole issue is that of
the isomorphism between the geometric and the algebraic systems under
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135
discussion. We will now describe it in more detail, while trying to avoid excessive rigor and formalism. The isomorphism consists of the well-known
structure-preserving bijective (that is, one-to-one and onto) map between
the geometric and the algebraic systems, introduced by René Descartes in
the 17th Century. To construct this map, one normally introduces, in the
plane, a pair of perpendicular axes with a unit on each and then assigns to
each point in the plane its coordinates (a pair of real numbers) relative to
the given axes. In addition, one assigns to each line in the plane a linear
equation ax + by + c = 0 (a and b not both 0) in such a way, that a given
point is on a given line if and only if the coordinates of the point are a
solution of the equation of the line. Using this map, each statement in one
of the systems has a unique ‘translation’ in the other system and it can be
shown that (with some caveats discussed below) a statement in one system
is true if and only if its translation in the other system is true.
It is absolutely essential for our purpose to note that this map depends
on the choice of a coordinate system – both the particular axes that we have
chosen arbitrarily and the units on each. Once we fix a coordinate system,
the map is uniquely determined, but for different systems we get different
maps. Thus, while the geometric and the algebraic systems are definitely
isomorphic, there are many isomorphisms (maps) between them, depending on the particular choice of coordinate system. More importantly, the
kind of statements whose truth-value is preserved under the isomorphism,
depends on the kind of the chosen coordinate system.
To explain this point more clearly, we need to introduce some terminology, specifically, to distinguish between geometric statements that do or
do not include the ‘metric’ notions of length, distance and angle. We will
call the system consisting of all the points and lines in the plane together
with the incidence relation (“a point is on a line”), but without the metric
notions, the restricted Euclidean plane5 . We will likewise call the extended
system, resulting when the metric notions of length, distance and angle are
added, the (full) Euclidean plane.6
Finally, we will denote algebraic objects corresponding to geometric
ones, by prefixing their names with the modifier ‘algebraic’. Thus, an
ordered pair of real numbers is an algebraic point, a linear equation7 an
algebraic
line and the function D, D((x1 , y1 ), (x2 , y2 )) =
(x2 − x1 )2 + (y2 − y1 )2 the algebraic distance between the algebraic
points (x1 , y1 ) and (x2 , y2 ). The system consisting of the set of algebraic points, the set of algebraic lines and the relation ‘is a solution of’
between them, will be called the reduced algebraic plane. When the algebraic distance function is added, the resulting system will be called the
(full) algebraic plane.
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Now, any legitimate coordinate system (that is, one made of an intersecting pair of axes – the condition required for just establishing the
bijective map) gives rise to an isomorphism between the Euclidean and
algebraic planes in the restricted sense. That is, it preserves the truth of any
statement that is equivalent to one formulated using only points, lines and
the incidence relation. Such isomorphism does not, in general, preserve
statements involving the metric notions of distance, length or angle. Metric statements are preserved if and only if the isomorphism is constructed
relative to a Cartesian coordinate system, that is, a homogeneous system
with the extra requirement that the algebraic length of the unit segment on
both axes is 1. Another way of saying this is that if we do not insist on
Cartesian systems, lengths and angles will vary for different systems and
will no longer be invariant.
It follows that once metric concepts are involved, we must work with
a Cartesian system; in a non-Cartesian system, the isomorphism between
the geometric and algebraic systems loses its utility, indeed its meaning,
since distances and angles, and the truth of statements involving them, are
not invariant anymore.8 Table IV summarizes the entities that are preserved
under the isomorphism induced by the various types of coordinate systems.
TABLE IV
Invariance of representations in different coordinate systems
Type of
coordinate system
Cartesian
Homogeneous
Non-homogeneous
Preserves
Incidence Slope Length
4.3. Resolution
In dealing with lines, their equations and their slopes in standard analytic
geometry, a homogeneous coordinate system is always implicitly assumed.
The occasions where this assumption is violated are for display purposes
only (viewing data on a logarithmic graphing paper or graphs on a computer display), but never in the actual course of mathematical work, such
as proof or problem solving. Under this implicit assumption, and the invariance of all the relevant concepts that it guarantees, we have justifiably
BEING SLOPPY ABOUT SLOPE
137
grown somewhat sloppy in treating as equal objects that correspond under
the isomorphism. For example, we talk about ‘the point (3,4)’ or ‘the slope
of the function y=2x+3’, instead of ‘the point whose coordinates are (3,4)’
and ‘the slope of the line representing the function y=2x+3’. In trying
to see how to deal with non-homogeneous coordinate systems, however,
we need to note carefully which terms are purely geometric and which
are purely algebraic and use each only within its own system, since its
representation in the other system does not remain invariant any more.
In particular, slope is a purely geometric term (a property of the position
of a line relative to the horizontal direction; no coordinate system involved)
and must be distinguished from the rate of change of a function.9 True,
relative to a homogeneous system, as noted earlier, the rate of change of
a linear function is equal to the slope of the line representing it, hence
the sloppy expression ‘the slope of the function’ makes sense; but this is
not the case any more in a non-homogeneous system. Here we can speak
meaningfully of the rate of change of the function and of the slope of the
line, but the two numbers are not related in any meaningful and consistent
manner.10 Returning to Task 2, what then is ‘the slope of the function’?
Our answer now is: the slope of the function (if you want to find it in
a geometric way) is the slope of any line representing the function in a
homogeneous system, thus must be 1. The ‘visual slope’, that is, the slope
of the line in the figure (a purely geometric entity) is indeed 1/3, but this
has no relation whatsoever (except perhaps in sign) to the ‘analytic slope’,
that is, the rate of change (or ‘slope’) of the function represented by this
line. We must find another, homogeneous, representation (e.g., that of Task
1) and calculate the slope of the line from that representation.
4.4. Comparing our mathematical analysis with the approaches found in
the study
We briefly re-examine the four approaches to Task 2, in the light of the
present mathematical analysis. How is each approach related to our analysis? According to approaches A, M1 and M2 , the slope of the function is
the same as the slope of a line representing it in a homogeneous system.
Thus, the slope in Task 2 is 1 – the same as in Task 1. Approach M2 is the
one most resonant with our analysis, since it distinguishes between angle
as a non-invariant geometric property of the graph and slope of a function
as an invariant. The approach least resonant with our analysis is approach
V , according to which slope is a purely geometric term, hence the slope of
a (linear) function is the slope of any line representing it. Thus, holders of
approach V maintained that the slope of the function in Task 1 is 1 while
the slope of the same function in Task 2 is 1/3.
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5. C ONCLUDING REMARKS
There are many explicit and implicit default assumptions and conventions
that we automatically make (individually or collectively) when learning
mathematics, mostly without being fully aware of these assumptions or
their consequences. What happens when we question these assumptions,
when we metaphorically “pull the rug from under the feet” of our students?
This is exactly what Tasks 1 and 2 set out to do: examine people’s behavior
when some specific default assumptions are violated. Choosing a simple
linear function to illustrate the problem turned out to be effective in this
respect. In Freudenthal’s (1983, p. 469) terms, Task 2 is a task that may
provide insight into a situation that has been ‘clogged by automatisms’.
The findings of this study, together with our mathematical analysis,
point to the value that this kind of exercise may have for students. Representing functions in different coordinate systems and the assumption
that “the behavior of the graphs of a function is invariant”, seem complicated enough issues that many individuals will need some help unpacking.
The tasks and the discussions that followed prompted the participants to
re-think and deepen their understanding of the underlying assumptions
beneath the coordinate systems in which they are accustomed to represent functions. The resulting disequilibrium provoked much discussion and
reflection and led to re-examination and refinement of the assumptions
concerning such basic notions as slope, scale, and angle. These considerations can be regarded as the kind of mathematical understanding that
Ma (1999) discusses with respect to teachers’ knowledge of mathematics.
On the other hand, given the amount of confusion and frustration that
these tasks sometimes caused, we tend to think that they may not be appropriate for the general high school population. They seem to us more
suitable for pre-service and in-service mathematics teachers and perhaps
for advanced high school students. We do not consider confusion and frustration during the learning process a bad thing in itself, but it is up to the
individual teacher to judge in each situation whether such emotional states
may help or hinder the students’ learning.
An examination of ten standard calculus textbooks and dictionaries
showed that many authors are prone to a bit of sloppiness in treating the
slope of a function. Most books we have checked defined the slope of a
(linear) function analytically, as difference-quotient or derivative. Some
books added an interpretation of the slope of the graph as angle of inclination (or its tangent), illustrating it in a homogeneous system or in a
coordinate system in which the units are not marked. Rarely do textbooks
refer to the issue of scale and the limitations of a non-homogeneous representation. But now, with the ubiquity of graphing software, these issues
BEING SLOPPY ABOUT SLOPE
139
cannot be avoided any more. As this paper demonstrates, the delicacy of
the mathematical issues involved requires that authors and teachers pay
explicit attention to them.
This research also suggests that interpreting ‘angle for slope’ in Rasslan
and Vinner’s (1995) terms should not be considered a misconception. The
angle between the graph and the axes in various situations is an important
aspect of slope. Teaching students to ignore it or treat it as irrelevant does
not contribute to understanding slope and may in fact lead to the kind of
confusion reported here. We suggest using any opportunity to enhance the
understanding of slope, by distinguishing between the ‘visual slope’ – the
slope of a line (for which the angle is a relevant feature) and the ‘analytic
slope’ – the rate of change of a function. Embedding such discussions in
the context of non-homogeneous change of scale can serve as a vehicle for
sharpening the understanding of slope as well as reflecting on the underlying assumptions and implications of representation of functions in various
coordinate systems (cf. Sela, 2000).
ACKNOWLEDGEMENT
We wish to thank Alla Shmukler for her contribution to the paper.
N OTES
1. For brevity, the two tasks are presented here together. In the actual research, Task 2
was given to the participants only after completion of Task 1. Thus the expressions
actually appearing in Question 3 in the two tasks were tan(α) and tan(β), respectively.
2. We adhere to the usual convention that the real numbers are ‘uniformly distributed’
along the line. That is, the difference between the two numbers that are represented by
the endpoints of any unit interval on the axis is fixed and independent of the position
of the interval on the axis.
3. We thank an anonymous reviewer for suggesting this view.
4. In thinking about these issues, we were strongly guided by standard approaches to
similar issues in linear algebra (in particular, the theory of Euclidean spaces). We have
strived, however, to keep our presentation free from linear algebra terminology.
5. In some countries this is called The affine plane.
6. Technically, it is enough to add a ‘distance function’, the notions of length and angle
being definable in terms of the distance function.
7. More precisely, an ‘equivalence class’ of linear equations, consisting of an equation
together with all its multiples by a nonzero real number.
8. Since the notion of slope of a linear function depends only on the ratio between
lengths, it will remain invariant in any homogeneous system.
9. As indeed is common in some countries.
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ORIT ZASLAVSKY ET AL.
10. Except that in practice we usually change scale by a positive factor, hence the two
numbers will have the same sign.
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Department of Education in Technology and Science,
Technion – Israel Institute of Technology,
Haifa, Israel,
Telephone +972-4-8262798, Fax +972-4-8325445,
E-mail: [email protected]