1999, Vol. 11, NO.2, pp. 124-144
Mathematics Education Research Journal
Secondary Mathematics Teachers'
Knowledge of Slope
Sheryl Stump
Ball State University
This study, conducted in the United States, investigated secondary mathematics
teachers' concept definitions, mathematical understanding, and pedagogical content
knowledge of slope. Surveys were collected from 18 preservice and 21 inservice
teachers; 8 teachers from each group were also interviewed. Geometric ratios
dominated teachers' concept definitions of slope. Problems involving the
recognition of parameters, the interpretation of graphs, and rate of change
challenged teachers' thinking. Teachers' descriptions of classroom instruction
included physical situations more often than functional situations. Results suggest
that mathematics teacher education programs need to specifically address slope as a
fundamental concept, emphasising its connection to the concept of function.
Current reform efforts for secondary mathematics education in several
countries revolve around several themes involving concepts and connections.
According to Curriculum and Evaluation Standards for School Mathematics (National
Council of Teachers of Mathematics [NCTM], 1989), "The proposed algebra
curriculum will move away from a tight focus on manipulative facility to include
a greater emphasis on conceptual understanding" (p. 150). Fey (1990) contends
that reformed algebra instruction should be guided by a small collection of
fundamental concepts such as variables, functions, relations, equations and
inequalities, and rates of change. Fey and Good (1985) also recommend that
secondary mathematics should emphasise functions. In addition, the NCTM
Standards document promotes the development of mathematical connections
within mathematics curricula so that students can recognise equivalent
representations of the same concept. Day (1995) suggests that functions may be
used as a unifying idea to make connections among mathematical concepts and
procedures, to make connections to other content areas, and to provide models of
real-world situations.
Despite these proposals for changes in the secondary mathematics
curriculum, a critical gap remains between recommendations and practice.
Although various reform curriculum materials have been developed, textbooks of
a more traditional nature still exist. Thus, an important link between the intended
curriculum and the implemented curriculum is teachers' knowledge: their
mathematical content knowledge and their pedagogical content knowledge
(Shulman, 1986). In order to provide some insight into the situation, this article
describes an investigation of secondary mathematics teachers' knowledge of the
concept of slope. The study examines teachers' mathematical understanding of
various representations of slope and their knowledge for teaching slope in order
to ascertain teachers' readiness to implement recommendations for a curriculum
that focuses on concepts, connections, and functions.
.
Secondary Mathematics Teachers' Knowledge of Slope
125
Conceptual Analysis of Slope
Slope is a fundamental mathematical concept in the high school curriculum.
In the United States, it is typically introduced in first-year algebra (Grade 8) and
then reappears in various forms throughout the secondary mathematics
curriculum. A conceptual understanding of slope is especially crucial for the
study of calculus or physics. Calculus typically begins with the study of
derivatives and rates of change, using slopes of lines to develop these concepts.
Physics assumes the ability of students to interpret slope as a functional
relationship between two quantities.
The slope of a line is a measure of its steepness. Algebra textbooks frequently
define the slope of a line as the ratio of the vertical rise to the horizontal run as
you move from one point to another along the line. Merriam-Webster's Collegiate
Dictionary (1993) mathematically defines slope as lithe tangent of the angle made
by a straight line with the x-axis" (p. 1106). These are geometric (and
trigonometric) notions. However, non-vertical lines are graphical representations
of linear functions, and we shift away from geometry when we consider alternate
representations of functions. For example, linear functions may take the forms
y = mx + b or ax + by = c when represented algebraically. In these cases we
represent slope with parameters as m or _!!.., respectively. We also use an
b
algebraic formula to calculate slope, Y2 - Y1 •
X 2 -Xl
Functions may appear as situations, verbal descriptions, tables, graphs, or
formulae (Janvier, 1987). Regardless of the form, the rate of change of a function is
the ratio of change in the dependent variable to the change in the independent
variable. This is the same ratio as is used to represent slope. Thus, although slope
has its reference in geometry, slope is rate of change and therefore it also has
meaning in formulae, tables, physical situations, and verbal descriptions.
Furthermore, slope is closely related to the concept of derivative. Slope is thus
considered a fundamental concept with many representations (e.g., geometric,
algebraic, trigonometric and functional). It is a deep mathematical idea that
threads its way throughout the curriculum, closely tied to thenotion of function.
This concept of slope challenges the distinction between ratio and rate.
According to Vergnaud (1983, 1988), a ratio is a comparison between quantities of
like nature and a rate is a comparison of quantities of unlike nature. In the case of
slope, this distinction becomes blurred depending on what variables the axes of a
graph represent. If they simply represent numbers, then slope is a ratio. However,
if they represent quantities of an unlike nature, such as distance and time, then
slope is a rate. The distinction lies in the situations in which the concept occurs.
Thompson (1994) focuses on the mental operations used to constitute ratio and
rate situations. A ratio is lithe result of comparing two quantities multiplicatively"
(p. 191), and a rate is " a reflectively abstracted constant ratio" (p. 192).
Proportional reasoning is an important mathematical prerequisite for the
concept of slope (Lamon, 1995). The complexities associated with children's
development of proportional reasoning have been well documented (Hart, 1981;
126
Stump
Heller, Post, Behr, & Lesh, 1990; Karplus, Pulos, & Stage, 1983; Singer & Resnick,
1992; Thompson & Thompson, 1994, 1996)._ Like Kaput and West (1994),
Thompson (1994) argues that rates involving time are the most intuitive, but time
as a quantity that can be imagined to vary proportionally with another quantity is
a non-trivial construction for students. Further abstraction is required to develop
an image of rate that entails covariation of two non-temporal quantities and the
average rate of change of some quantity over some range of an independent
quantity. This abstraction may be difficult even for preservice teachers (Simon &
Blume, 1994).
The complexity of the concept of slope is further complicated by the various
common meanings associated with the word "slope". Merriam-Webster's Collegiate
Dictionary (1993) indicates that, as a noun, slope is an upward or downward slant
or inclination; ground that forms a natural or artificial incline; or the part of a
continent draining to a particular ocean. As a verb, to slope is to take an oblique
course, to lie or fall in a slant, or to cause to incline or slant.
Students' and Teachers' Knowledge of Slope
Research has documented difficulties that students have with the concept of
slope. There are misconceptions associated with the calculation of slope (Barr,
1980, 1981) and with the interpretation of linear functions and their graphs (Barr,
1980, 1981; Moschkovich, 1990; Schoenfeld, Smith, & Arcavi, 1993). Students also
have difficulty relating graphs to linear equations (Kerslake, 1981) and to the
notion of rate of change (Bell & Janvier, 1981; Janvier, 1981; McDermott,
Rosenquist, & Van Zee, 1987; Orton, 1984). They have difficulty considering slope
as a ratio (Barr, 1980, 1981) and connecting the ratio to a physical model (Simon &
Blume, 1994). In a study in Spain, Azcarate (1992) found that two months prior to
the study of derivatives, confusion about the definition of slope existed among
high school calculus students.
No research has been found that directly addresses teachers' knowledge of
slope. However, several investigations (Even, 1993; Norman, 1992; Stein, Baxter,
& Leinhardt, 1990; Wilson, 1994) have explored various aspects of preservice and
inservice teachers' knowledge of function. These studies found that, for the most
part, teachers have limited knowledge of this fundamental concept and that their
level of understanding is reflected in their teaching. In contrast, Lloyd and Wilson
(1998) observed that an experienced high school teacher's rich conceptions of
functions contributed to his skillful implementation of a reform curriculum.
Theoretical Perspective
This investigation was guided by the complexity of the concept of slope. It
considered a collection of various representations of slope as a lens to focus on
three aspects of teachers' knowledge of slope: their concept definitions, their
mathematical understanding, and their pedagogical content knowledge.
Secondary Mathematics Teachers' Knowledge ofSlope
127
Concept Definitions
The theory of concept images and concept definitions (Tall & Vinner, 1981) is
useful in analysing teachers' knowledge of slope. A' concept image is the total
cognitive structure that is associated with a given concept. A concept definition is
a set of words used to specify a given concept. This investigation directly
examined teachers' concept definitions of slope. Teachers' concept images of slope
were inferted in their comments about mathematics instruction.
Mathematical Understanding
Flexible understanding of mathematics entails the ability to form relationships
within mathematics as well as across disciplinary fields and to make connections
to the world outside school (McDiarmid, Ball, & Anderson, 1989, p. 193).
Meaningful learning of mathematics includes forming relationships between
conceptual and procedural knowledge (Hiebert & Lefevre, 1986). Conceptual
. knowledge is knowledge that is rich in relationships, linking new ideas to ideas
that are already understood. Procedural knowledge consists of formal language
and symbol systems as well as algorithms and rules. A flexible understanding of
slope includes conceptual knowledge of various representations of slope. It also
includes procedural knowledge of the symbols typically used in relation to slope
(for example, m and
~y) and the rules used to calculate slope.
~x
Pedagogical Content Knowledge
Shulman (1986) defined pedagogical content knowledge as lithe ways of
representing and formulating the subject that make it comprehensible to others!'
(p. 9). The category includes knowledge of representations, analogies,
illustrations, examples, explanations, and demonstrations for particular topics. An
understanding of the potential misconceptions of students associated with specific
topics is also part of pedagogical content knowledge.
It is important that secondary mathematics teachers have a robust knowledge
in order to help their students develop a' powerful understanding of slope. The
present study looked for evidence that teachers know slope as a fundamental
concept with various representations. In particular, it searched for evidence that
teachers appreciate the connection between the concepts of slope and function,
and the notion of slope as rate of change. The following research questions guided
the investigation: What representations of slope are included in teachers' concept
definitions? Do teachers have a flexible understanding of various representations
of slope? What representations of slope are included in teachers' descriptions of
classroom instruction? What real-world situations do teachers use to illustrate the
concept of slope? What is their knowledge of students' difficulties with slope?
How does preservice teachers' knowledge of slope compare with the knowledge
of inservice teachers?
128
Stump
Method·
Participants and Data Collection
In this article, the term "teachers" represents both preservice and inservice
teachers. Eighteen preservice participants (10 males and 8 females) were
volunteers enrolled in a secondary mathematics methods course one semester
prior to student teaching. Twenty-one inservice participants (12 males and 9
females) were volunteers from four high schools in central Illinois. All but 2 of the
inservice teachers had bachelor degrees in mathematics or mathematics
education. Of the 15 inservice teachers who had masters degrees, 8 had degrees in
either mathematics or mathematics education. Their teaching experience ranged
from 1 to 32 years. All but one inservice teacher had taught Algebra I (Grade 9);
all but two had taught Algebra II (Grade 11); all but one had taught Geometry
(Grade 10); all but five had taught Trigonometry (Grade 11 or 12); six had taught
Calculus (Grade 12); and two had taught Statistics (Grade 12).
All participants, both preservice and inservice teachers, completed a survey
questionnaire. From the sets of those teachers willing to be interviewed, a random
subset of four females and four males from each group was selected for interview.
Instruments
The mathematics survey and the interview protocol contained questions
about teachers' definitions of slope, mathematical problems involving various
representations of slope, and questions about mathematics pedagogy. The survey
contained 11 items. The length of the interviews, which were audio-recorded and
later transcribed for analysis, varied from about 45 minutes to one hour. The
survey and interview questions are summarised in the results sections below.
Copies of the instruments are available from the author.
Data Analysis
On the mathematics survey. and the interview transcripts, teachers' responses
to questions involving definitions and mathematics pedagogy were coded
according to the types of mathematical representations included. Responses to
questions assessing mathematical understanding were checked for correctness
and coded according to the types of mathematical representations employed to
solve the problems.
The transcripts were then analysed using the process of analytic induction
described by Erickson (1986). After repeated readings of the data, various
assertions were. generated. The validity of those assertions was then tested by
reviewing the transcripts repeatedly, seeking confirming as well as disconfirming
evidence. Patterns were revealed. through analysis within individual teachers'
responses as well as through comparisons among various teachers' responses.
Secondary Mathematics Teachers' Knowledge ofSlope
129
Results of the Mathematics Survey
The results of the mathematics survey provide a broad picture of secondary
mathematics teachers' knowledge of the concept of slope. They indicate general
trendsconceming the representations of slope included in teachers' definitions as
well as teachers' mathematical understanding of slope. Furthermore, they provide
a general portrait of their descriptions of classroom instruction.
Teachers' Definitions of Slope: Categories of Representations
On the mathematics survey, two questions probed teachers' concept
definitions of slope: "What is slope?" and "What does slope represent?" The
purpose of the second question was to encourage teachers to think further about
the meaning of slope. The representations of slope included in teachers'
definitions were sorted into seven categories as follows.
.
.
ri~
• The category of geometrzc ratIO included representations such as "run "
•
and "vertical change over horizontal change" and focused on slope as a
geometric property.
The category of algebraic ratio included representations such as "Yz - Yl "
-X2
•
•
•
•
•
-Xl
and "the change in y over the change in x", in which slope was defined by
an algebraic formula.
The words "slant", "steepness", "incline", pitch", and "angle" were
categorised as involving a physical property.
Responses referring to slope as the rate of change between two variables
were categorised as involving afunctional property.
The parametric coefficient category included references to m in the equation
y = mx + b.
A trigonometric conception of slope referred to the tangent of the angle of
inclination.
A calculus conception included mention of the concept of derivative.
In their responses to the two questions about concept definitions, the mean
number of different representations per teacher was 2.6 for preservice teachers
and 2.7 for inservice teachers. As shown in Table I, geometric ratios dominated
both preservice and inservice teachers' definitions of slope. Inservice teachers
mentioned physical properties more often than preservice teachers. Preservice
teachers mentioned algebraic ratios and functional properties more often than
inservice teachers. Only two teachers defined slope as a parametric coefficient, the
same number named a trigonometric conception, and only three teachers named a
calculus conception.
130
Stump
Table 1
Percentages of Preservice and lnservice Teachers
Slope in Their Concept Definitions
Representation of Slope
ln~luding
Preservice (n = 18)
Various Representations of
Inservice (n
Geometric ratio
83
86
Algebraic ratio
61
57
Physical property
44
81
Functional property
50
33
Parametric coefficient
11
o
Trigonometric conception
o
10
Calculus conception
6
10
=21)
Mathematical Understanding of Slope
The collection of problems on the mathematics survey, shown in Figure I, was
designed to include a variety of representations of slope. In the various items,
slope appears as a geometric property of a line, a parameter of a linear equation, a
rate of change in a functional situation, and a trigonometric concept in relation to
an angle of inclination. (Note: The headings in italics were not included when the
problems were presented to the teachers.) Table 2 shows teachers' success rates on
these problems.
Table 2
Preservice and lnservice Teachers' Percentage Success Rates on Mathematics Problems
Problems
Preservice
(n = 18)
Inservice
(n = 21)
1. Rate of growth
100
100
2. Equation with numerical parameters
100
100
3. Cost per additional minute
Correct slope
Correct interpretation
89
89
100
90
4. Slope as speed
Part A
Part B
78
67
76
76
5. Equation with literal parameters
Correct equation
Correct expression for slope
Correct expression for y-intercept
83
61
33
81
62
57
6. Angle of inclination
33
67
Secondary Mathematics Teachers' Knowledge ofSlope
131
1. Rate ofgrowth. At age 9 Karen was 4'3" tall. At age 11 she was 4'9" tall. How
fast did she grow from age 9 to age II? (McConnell, Brown, Eddins,
Hackworth, & Usiskin, 1990)
2. Equation with numerical parameters. A student takes a test and gets a score of
50. She gets a chance to take the test again. It is estimated that every hour of
studying will increase her score by 3 points. Let x be the number of hours
studied and y be her score. Find an equation which represents this $ituation.
(McConnell et al., 1990)
3. Cost per additional minute. The following two points give information about an
overseas telephone call: (5 minutes, $5.30), (10 minutes, $10.80).
A. Calculate the slope.
B. Describe what the slope stands for in the context of this situation.
(McConnell et al., 1990)
4. Slope as speed. The figure below shows a position versus time graph for the
motions of two objects A and B that are moving along the same meter stick.
P
A
,...-.,
S
.~
~
0
'+}
';;:: 20
0
CL.o
0
A.
B.
123456789
Time (sec)
t
At the instant when t = 2 sec, is the speed of the object A greater than,
less than, or equal to the speed of object B? Explain your reasoning.
Do objects A and B ever have the same speed? If so, at what times?
Explain your reasoning. (McDermott, Rosenquist, & Van Zee, 1981)
5. Equation with literal parameters. For small changes in temperature, the formula
for the expansion of a metal rod under a change in temperature is 1 - 10 =
a 10 (t - to), where 1is the length of the object at a temperature t, 10 is the initial
length at temperature to, and a is a constant which depends on the type of
metal.
A. Express 1as a linear function of t.
B. Find the slope and the y-intercept. (Hughes-Hallett et al., 1992)
6. Angle of inclination. A line makes an angle of 60° with the x-axis and passes
through the point (3, 1). Is it possible to find the slope of this line? If yes,
what is the slope? If no, why is it impossible to find?
Figure 1. Problems on th~ mathematics survey.
132
Stump
All of the teachers, both preservice and inservice, correctly answered the first
two problems (Rate of growth and Equation with numerical parameters). All the
teachers except for two preservice teachers correctly calculated the slope in Item 3
(Cost per additional minute), but their interpretations varied. Moschkovich,
Schoenfeld, and Arcavi's (1994) distinction between the process perspective and
the object perspective in relation to linear relations was used to analyse teachers'
interpretations of the slope. Three preservice teachers and three inservice teachers
interpreted the slope as a process, focusing on the relationship between the values
of the variables. For example, two interpretations of the slope as a process were,
"the change in cost proportional to the change in time" and" change in price The
change in time
remaining teachers interpreted the slope as an object. From the object perspective,
according to seven preservice teachers and twelve inservice teachers, slope
represented the cost per minute of a phone call. However, five preservice teachers
and six inservice teachers were more careful in their in interpretations of slope as
an object. For example, one inservice teacher wrote, "If 5 minutes for $5.30 is the
minimum charge, the charge for additional time is $1.10 per minute."
In Item 4 (Slope as speed), four preservice teachers and three inservice
teachers said the speed of Object A was less than the speed of Object B. Two
preservice teachers and one inservice teacher answered Part A correctly, but
indicated in Part B that the two objects would have the same speed at some point.
Two inservice teachers left the question blank. The second part of Item 5
(Equation with literal parameters) provided difficulties for both sets of teachers.
Inservice teachers outperformed preservice teachers on Item 6 (Angle of
inclination). A typical incorrect response was the following: "We need another
point in order to determine the change in y over the change in x."
If.
Pedagogical Content Knowledge
Prerequisites for students' understanding. Teachers responded to the following
question on the mathematics survey: "What mathematical concepts must students
have experience with before they can truly understand slope?" Table 3 shows that
both preservice and inservice teachers mentioned geometric concepts and skills
most often in their responses. The Cartesian coordinate system, plotting points,
and graphing were among the geometric concepts and skills identified by both
preservice and inservice teachers. Variables, formulae, and solving equations
were among the algebraic concepts and skills mentioned. Graphing linear
equations was categorised as both geometric and algebraic. Both preservice and
inservice teachers listed arithmetic concepts such as fractions, addition, and
subtraction. Inservice teachers additionally included the following arithmetic
concepts: percentages, positive and negative numbers, and division involving
zero. Only three teachers from each group mentioned the importance of
knowledge of functional relationships.
I
Secondary Mathematics Teachers' Knowledge ofSlope
133
Table 3
Percentages of Preservice and Inservice Teachers Identifying Various Concepts and Skills
as Prerequisites for Students' Understanding of Slope
Concepts and Skills
Preservice (n = 18)
Inservice (n = 21)
Geometric
100
81
Algebraic
44
33
Arithmetic
33
43
Functional
17
14
Ratio
11
19
Real-world situations involving slope. The mathematics survey provided
information about the types of instructional representations teachers thought
about in relation to the concept of slope. First, teachers provided examples to
illustrate" their definitions of slope. Second, they answered the question, "What
analogies, illustrations, examples, or explanations do you think are most useful or
helpful for teaching the concept of slope?" Teachers mentioned standard formulae
and graphs, but they also mentioned real-world situations involving slope. These
situations fell into two categories, referred to as physical and functional. Physical
situations focused on the steepness of physical objects such as mountain roads, ski
slopes, or wheelchair. ramps. Functional situations emphasised the linear
relationship between two varying quantities such as distance and time.
Table 4
Percentages of Preservice and Inservice Teachers Mentioning Various Types of Situations
Involving Slope on the Mathematics Survey
Types of Situations
Preservice (n = 18)
Inservice (n = 21)
Physical
39
62
Functional
22
9
Both
11
5
Neither
28
24
As shown in Table 4, both preservice teachers and inservice teachers were
more likely to mention physical situations than functional situations. Physical
situations were mentioned by more inservice teachers and functional situations
were mentioned by more preservice teachers. About a quarter of both· the
preservice and the inservice teachers did not mention any real-world examples.
Results of the Interviews
The data obtained from the interviews provide an extended view of
preservice and inservice teachers' mathematical understanding of slope and their
134
Stump
pedagogical content knowledge of slope. Teachers worked to solve additional
problems involving various representations of _slope, and they discussed their
ideas about teaching slope. In the following, the teachers' actual names are
replaced with pseudonyms. First names are assigned to preservice teachers, and
last names are assigned to inservice teachers. To save space, names are sometimes
replaced by initials.
Suppose that, using a meter stick, we measure the diameters and the
circumferences of several aluminium disks. If we plot a graph of our data with
diameter (in cm) on the x-axis and circumference (in cm) on the y-axis, will the
graph be a line? What is the slope of this line?
8~~
u
'-"
Diameter (em)
-
Figure 2. Slope as a functional relationship and a parameter.
Teachers Solving Various Slope Problems
Slope as afunctional relationship and a parameter. The problem shown in Figure 2
involves the functional relationship between the circumference and the diameter
of a circle. Five of the eight preservice teachers (one did not answer the question)
and seven of the eight inservice teachers successfully solved this problem. Of
these, four preservice teachers and six inservice teachers relied only on their
examination of the linear equation c = pd to determine that the graph would be a
line and to identify the parameter p as the slope of the line.
Mr. Eno described the connections that he made as he solved the problem:
Mr. E:
We're plotting the circumference versus the diameter. Yes, it will [be a
line]. And the reason it will be a line is because the equation is c = pd +
0, which indicates it will go through the origin, and your slope should
be p, because that would be your mx + b. If this was in our Algebra II
book, we would even say that was a direct variation. Because if you
have a line that passes through the origin, then we call that direct
variation, where the slope is m and the y-intercept is zero. And we just
say the circumference would vary directly with the diameter.
As a supplement to their successful analyses of the linear equation, Francine
and Ms. Hall constructed a graph in order to examine the situation geometrically.
Francine plotted points and sketched a graph, but evidently she did not use the
Secondary Mathematics Teachers' Knowledge ofSlope
135
graph to determine the slope:
OK, circumference is equal to p times diameter, so taking p to be 3, I
found the points. I said if the diameter is 2, the circumference is 6. And
if the diameter is 3, then the circumference would be 9. I guess I would
have to find another point to be sure. If the diameter is 4, then the
circumference would be 12. Yes, it is a line.
I (Interviewer):: Do we know anything more about the line? Do we know what
the slope is?
F:
Slope is going to be p.
F:
Ms. Hall described how she plotted points to determine the graph was a line and
how she used the equation to determine the slope:
Ms. H:
I:
Ms. H:
I:
Ms. H:
It should have a line with a slope of p.
Why do you 'say that?
I started picking some examples so that if my diameter was 1, then the
circumference is p. If my diameter is 2, then circumference is 2p. So you
end up with a line.
How did you decide that the slope was p? What were you looking at to
make that decision?
I was plugging the diameter and the circumference into the equation for
a line. So the slope and p had to be the same thing.
However, those who did not successfully solve the problem (Andy, Holly,
and Ms. Ball) did not use the linear equation to determine the slope. Andy used
the equation only to plot points, and his attention then remained on the graph. He
relied on the appearance of his graph and thus was unsuccessful in his .
determination of the slope of the line:
A:
I:
A:
Well [pause]. I believe [the slope] would be l.
What is making you think I?
Because initially if you had the diameter equal to 1, circumference is just
equal to p. But if you double that distance, and make it 2, you have 2p.
No, I'm wrong. I don't want to say that, either. What I am trying to do is
get a graphical representation and see what the line looks like. I'm
trying to get these to scale, and that is kind of hard to do. So [graphing].
Yes, I would say the slope is l.
Holly used the equation to construct a table which she then used to examine
the relationship between the variables c and d:
I:
H:
I:
H:
I:
H:
Would the graph be a line?
So you have 2pr. You're increasing it p. I would say yeah, it would be.
What do you know about the line? Do you know the slope of the line?
I would guess around p.
Why would you guess around p?
Well, this [the column of her table containing the values for d] is
increasing. This [the d column] is p times larger than this [the c column].
So for every diameter, your circumference is p times that. I'm not really
sure if it would be the slope, but I'm thinking it would be related to it.
Ms. Ball drew pictures of
t~o
circles, one with a diameter of 10 and one with
136
Stump
a diameter of 20. She calculated the circumferences of the two circles, and then she
drew a graph 'of a line using the points (10,. lOp) and (20, 20p). She then
determined that the slope of the line was 2 by calculating
~O_- 1~0
. While focusing
on a formula for calculating slope (which she used incorrectly), she lost sight of
the connections between her graph, the functional relationship, and the linear
equation.
Slope as rate of increase. The problem shown in Figure 3 (Orton, 1983) probed
teachers' knowledge of slope as a rate of increase. All of the teachers answered
parts A and B correctly (except for one inservice teacher who did not answer the
question due to time restrictions). Six preservice teachers and seven inservice
teachers answered part C correctly. However, only four preservice teachers and
five inservice teachers provided correct responses to parts D and E. These
teachers demonstrated understanding of the notions of increase and rate of
increase and, in most cases, they used 'the word "slope" to describe how they
solved the problem.
y = 3x - 1
-
/
A.
B.
C.
D.
E.
x
/
What is the value of y when x = a?
What is the value of y when x = a + h?
What is the increase in y as x increases from a to a + h?
What is the rate of increase of y as x increases from a to a + h?
What is the rate of increase of y at x =2 l ?
2
Figure 3. Slope as rate of increase.
George's explanation revealed
understanding.
I:
G:
the
connections in his
mathematical
Can you tell me how you got 3 for part D?
Well, 3 is the rate, 3 is the slope of the line. So that's the rate that it's
increasing, regardless of how much h is. This is alinear equation, so the
rate is a constant. If it weren't linear, you'd be doing an approximation
of the rate of increase. But, since it is a linear equation, you're not doing
an approximation. You're getting an exact rate. And so it's 3. Another
way to think of it would be that the rise would .pe, the change in y
would
be
3h,
and the change in the x-value would be h. So it's 3h over h, which is the
Secondary Mathematics Teachers' Knowledge ofSlope
same as 3.
137
.
Mr. Car explained how he applied his understanding of slope to successfully
solve the problem in this way:
Mr. C:
Part 0, I took the ratio of change in y to change in x. And then realised
that I didn't need to do that, because I understand the concept of slope. I
should have know that the rate of increase was 3. And then similarly for
part E, the rate of increase is constant. So I knew that was also 3.
The four preservice teachers and two inservice teachers who were not
successful with the problem seemed to lack understanding of rate of increase in
this situation or they failed to see the relationship between rate of increase and the
equation of the line. For example, Diana explained her thinking and revealed her
misconception about rate of increase.
0:
In C, they are asking for a number or a letter increase. In 0, they are
asking for rate, so that's like a fractional or a decimal-type thing. Rate is
a proportion. And actual increase, they are just asking for a single
figure.... I think that ~ increases by 3h. And the rate of increase would
be like rise over run.... [Writes a +h.] ... I'm not sure that E has an
h
answer.... There is only one point. For the rate of increase, you need to
have a comparison from one point to the other. There is no rate of
increase at one point.
Calvin also struggled with the notion of rate of increase. He knew there
was a relationship with slope, but he failed to connect rate of increase with the
slope of this particular line.
C:
I:
C:
I:
C:
I:
C:
[Writes 3h for C.]
How did you get 3h?
Maybe that would have been the rate of increase? What is the increase?
Weil, it's 3h. When I multiply that out, I'll just subtract it. What is the
rate of increase? Hmm. Well, it's h, isn't it?
How do you get h?
I don't know. I can't really answer that one to tell you the truth. [Looks
at E.] It's zero. Either that or it would be undefined. Yeah, it's
undefined. [Goes back to D.] I hope I'm not the only one that had
trouble with this?
What does that mean, rate of increase?
Well, from right here it was the slope. I guess that's what it is. That's
what it was here. So rate of increase would be .... Slope here was .., . I
don't know. I'm sorry.
The trigonometric representation of slope. The questions shown in Figure 4 were
designed to determine if teachers connected the slope of a line to the tangent of
the angle of inclination.
All but two of the inservice teachers, Ms. Ball and Ms. Fell, acknowledged the
connection between slope and the tangent function. When responding to the first
question in Figure 4, Ms. Day immediately made a connection between the
geometric and the trigonomet!ic representations of slope but suggested that a
138
Stump
student who makes this connection has greater understanding than a typical
beginning algebra student:
1. A student states that the slope of a line is related to the angle formed by the
line and the x-axis. How would you respond?
2. Line 1forms an angle of measure 8 with the x-axis, and line k forms an angle
of measure 28 with the x-axis. A student states that the slope of line k is twice
the slope of line 1. How would you respond?
y
k
Figure 4. The trigonometric representation of slope.
Ms. D:
I'd move him into an advanced honours class first thing, because it
would take more of a discussion than that. First, we would have to see
where he thinks this angle is being formed with the x-axis. Then I would
want to know what connections he's making with that, how he thinks
that's related. Because if he's moving on into tangent, then you have
someone who doesn't belong in their math class. But, yeah, there is a
relationship.
Only two of the eight preservice teachers, Brenda and George, provided
correct responses to both questions involving the trigonometric representation of
slope. George made no specific reference to the word "tangent", but pointed out
the benefit of having two ways of looking at the concept of slope:
G:
I think that would be a really good opportunity to show that there is a
different way of looking at how steep a line or something is. You can
look at the angle. That tells you how steep it is also. And to see that
sometimes one representation may be more useful than the other, but
that you can convert between them.
Some teachers identified a direct relationship between the angle and the slope.
Calvin and Ms. Ball expressed uncertainty about being able to determine the
actual slope of the line if they knew the measure of the angle.
C:
I don't know. I haven't really been taught that. I honestly haven't.
Maybe I don't remember, or I don't know. I'm sure there is a
Secondary Mathematics Teachers' Knowledge ofSlope
Ms. B:
139
relationship.
So if it's a positive slope, it would be an acute angle with the x-axis. If
it's a negative slope, it would have an obtuse angle with the x-axis. So
there probably is a relationship, but I've never thought of it before.
Holly expressed a similar uncertainty about responding to the first question in
Figure 4:
H:
If I had the equations of the line it would be helpful. I'm not really sure.
To tell you the truth, I wouldn't really know if he was correct without
the equation of the line to analyse.... I've never seen anything like this.
When I learned slope, we never talked about the angle.
Teachers Discussing Mathematical Instruction
The interviews provided opportunities for teachers to discuss their
knowledge of students' difficulties with slope. Inservice teachers spoke about the
actual difficulties they had witnessed during their years of experience with
students. They echoed the difficulties with symbolic interpretation and
manipulation documented in the literature on students' understanding of slope
(Barr, 1980, 1981; Schoenfeld, Smith, & Arcavi, 1993). Almost all the inservice
teachers mentioned students' confusion between rise and run or between x and y
in the formula for slope:
Ms. B:
Mr. C:
Well, if they don't know the formula, one main thing, they put the x's
over the y's. They do that all the time.
Reversing the numerator and the denominator. In terms of the
definition, change in y over change in x. ... And I would say the order in
which they subtract them. It takes a little bit of time to get used to the
idea that you can do it either way as long as we do it in the same order.
Preservice teachers hypothesised potential difficulties. They also mentioned
difficulties with symbolic manipulation:
C:
E:
Possibly positive and negative slope. The term, " m equals". At first I
was like, m? Just the variable m. When you're graphing, if the slope is 3,
you might do 3 x's and 1 y. They might have difficulties getting that
figured out.
I think the no slope for vertical lines would probably be a big one. Why
does a vertical line not have a slope, and a horizontal one does? And
then there's going to be confusion probably between positive and
negative.
Both preservice and inservice teachers expressed concern with students
understanding the meaning of slope.
D:
F:
G:
I think seeing the big picture.... I don't think it's difficult to find m. I
. think it's a really basic- it's basic arithmetic. But I think actually being
able to go further with slope, seeing that it does mean something.
I think a lot of times students can maybe calculate it but they don't
know what it really means, what they are calculating.
My guess is that some might be frightened off as soon as you introduce
a mathematical definition or a formula for a line, like the slope-intercept
140
Stump
H:
Mr. C:
Mr. G:
of the equation. As soon as some people see equations, they just go nuts,
especially with symbols instead of numbers.... Not because they don't
und,erstand what slope is, but because they are not making the
connection between the intuitive and even the not-so-intuitive idea of
taking the ratio of this to this. Not making the connection between that
and the symbolic abstract equation on paper. That's just a guess. I
haven't had experience with that.
If they didn't know what an unknown represented, that there could be
several different values of x in this equation that would work. .,. If they
didn't really understand that there could be different values for x, then I
don't think they would understand that the line is changing, that's what
the slope is, the rate of change of the y over the x.
[I want students to know] what it means and how it would apply in a
mathematical sense, to solve common problems or at least to relate to
common problems.
Just in general, what does it mean? When you say this line has a slope of
3
2' what does that mean? What does the 3 mean? What does the 2 mean?
Discussion
As evidenced by their definitions of slope, a substantial majority of the
preservice and inservice teachers in this study thought of slope as a geometric
ratio. A majority of the inservice teachers also described slope as a physical
property. A very small proportion, less than a fifth of each group, thought of
slope as a functional concept. In other words, the majority of teachers did not
think of slope as a rate of change between two variables. These findings are
comparable to those of Azcarate (1992), who investigated high school students'
definitions of slope. The majority of students in her study used geometric
language to define slope, but the use of functional language was much less
prevalent. Although teachers in this investigation may have been capable of
making connections between the concepts of slope and function, few incorporated
these connections into their expressed concept definitions.
Livingston and Borko (1990) found differences in the teaching practices of
pteservice and inservice teachers and hypothesised that preservice teachers were
hindered by a lack of connections in their content, knowledge. Indeed, the
inservice teachers in this study showed a better understanding of the connection
between slope and angle of inclination. Perhaps that is because so many of the
experienced teachers had taught trigonometry. Additionally, the study of
trigonometry is not typically part of undergraduate mathematics teacher
preparation in the United States.
In relation to their success with problems involving parametric and functional
representations of slope, the inservice teachers in this study demonstrated no
advantage over the preservice teachers. Some teachers from both groups had
difficulties with tasks involving the recognition of parameters, the interpretation
of graphs, and the recognition of slope as rate of change. These tasks are not
typical high school textbook problems, but the level of mathematics involved is
not beyond high school mathematics. They are tasks that require the utilisation of
connections among various representations' of slope. Teachers who recognised
Secondary Mathematics Teachers' Knowledge of Slope
141
and used those connections were more successful with the mathematical tasks in
the present investigation.
Considering the importance of the study of functions for high school students,
it is especially troubling that functional situations involving slope were missing
from so many teachers' descriptions of their instructional practices. Their students
may thus miss opportunities to make this important connection while forming
their conceptions of slope. Rizzuti (1991) found that instruction that included
multiple representations of functions allowed students to develop comprehensive
and multi-faceted conceptions of functions. Based on the results of the present
investigation, it is questionable whether the participating teachers could assist
their students in developing such a rich conception of slope.
Both preservice and inservice teachers considered physical situations
involving slope to be most useful for instruction. They used ramps, ski slopes, and
mountain roads as analogies for slope. However, none of the teachers mentioned
physical representations as a prerequisite for understanding the concept of slope.
Instead they named geometric, arithmetic, algebraic, and functional concepts.
According to Simon and Blume (1994), students may have trouble making the
connection between real-world situations and the mathematical expressions used
to represent quantitative relationships. A question for further investigation is
whether the students of the teachers in this investigation understand the
connections between physical representations and other representations of slope.
The present investigation was not designed to gather evidence to determine if the
teachers themselves understand the connections.
Ironically, both groups of teachers expressed concern about students'
understanding of the meaning of slope. However, the specific student difficulties
they identified focused on procedural aspects and reflected an emphasis on
standard algebraic and geometric representations of slope-reflecting the
teachers' own concept images of slope. Other research shows how a teacher's
instructional focus may be constrained by his or her own mathematical
conceptions (Lloyd & Wilson, 1998; Stein, Baxter, & Leinhardt, 1990). The present
study contributes to our understanding by revealing a variety of representations
in teachers' conceptions of slope-suggesting that instruction may vary in
emphasis and be limited by teachers' knowledge of the various representations.
The curriculum resources available to both preservice teachers and inservice
teachers may affect their mathematical understanding of slope and their
pedagogical content knowledge of slope. If their textbooks present slope, linear
equations, rate of change, and trigonometric functions as unrelated concepts, then
teachers may teach them as unrelated concepts. Some important questions to
investigate are the following: What connections do textbooks make between
various representations of slope? When textbooks connect various representations
of slope, do teachers emphasise those connections for their students? Can teachers
learn to make connections even if textbooks do not emphasise them? Can they
learn to make connections from their experiences in formal teacher education?
We cannot assume that teachers will make connections among various
representations of slope on their own. The results of the present investigation
suggest that the education of secondary mathematics teachers needs to specifically
]42
Stump
address various representations of slope (among other fundamental mathematical
concepts contained in the secondary mathematic? curriculum). According to the
Mathematical Association of America, "The mathematical preparation of teachers
must provide experiences in which they develop an understanding of the
interrelationships within mathematics and an appreciation of its unity"
(Mathematical Association of America, 1991, p. 3).
Cooney (1994, p. 17) suggested that "teachers' mathematical experiences must
also have a reflective and adaptive orientation", providing opportunities for
teachers to reflect on their experiences as learners of mathematics, on mathematics
itself, and on the relationship of mathematics to the real world. In particular, both
preservice and inservice mathematics teachers need opportunities to examine the
concept of slope, to reflect on its definition, to construct connections among its
various representations, and to investigate functional situations involving
physical slope situations.
. Acknowledgment
The research reported in this paper is part of the author's doctoral
dissertation (Stump, 1996) completed at Illinois State University under the
direction of Jane O. Swafford.
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Author
Sheryl L. Stump, Department of Mathematical Sciences, Ball State University, Muncie, IN
47306-0490. E-mail: <[email protected]>.