Running head: SLOPE: RATE OF CHANGE AND STEEPNESS
Slope as a Rate of Change and Steepness:
Definition, Misunderstandings and Applications
Pamela R. Hudson Bailey
George Mason University
EDCI 855: Mathematics Education Research on Teaching and Learning
Slope as a Rate of Change and Steepness:
Definition, Misunderstandings and Applications
The study of rate of change and steepness of a line begins in middle school and continues through Algebra I and II, Geometry, Trigonometry and Calculus (Anderson & Nelson, 1994). Science courses tie their success to a student’s ability at interpreting and constructing graphs thereby linking mathematics to another discipline (Berg & Smith, 1994). Farenga and Ness (2005) posit that one “…cannot underestimate the significance of rate…” (p. 58). With the significance placed on rate of change, its definition and misunderstandings need to be understood by teachers so that students might gain conceptual understanding of the topic. Misunderstandings include concerns with formulas and relating slope to physical models. Multiple representations and applications of slope, technological applications, writing slope as a fraction, decimal or percent, and teacher issues are also discussed.
Slope Defined
“What is slope?” A question asked or analyzed by students, pre-service teachers, in-service teachers and mathematics educators. The concept of slope takes on many different meanings with teachers and students. Descriptions and classifications include categories such as the physical and functional (Stump, 2001a, 2001b) and the analytic and visual (Zaslavsky, Sela & Leron, 2002). Further, slope may be defined as implementing repeated addition and proportional relationships (Walter & Gerson, 2007). A focused explanation of slope as a rate of change includes the categories of instantaneous, average and constant (Farenga & Ness, 2005). Each of the above categories overlaps and intertwines as different viewpoints define slope.
Six mathematics teaching majors were the focus of one of the studies by Stump (2001a) as a practitioner research approach was taken to understanding slope. The second study by Stump (2001b) involved 22 high school students taking pre-calculus examining their conceptual knowledge about slope. Algebraic, geometric, trigonometric and calculus interpretations are discussed by Stump (2001b). Algebraically slope is the difference of the dependent values divided by the difference in the corresponding independent values, geometrically it is the steepness of a line or the rise divided by run, the tangent of the angle is the trigonometric interpretation while in calculus slope is viewed as a limit of a function (Stump, 2001a). Overall slope is categorized into two different areas, physical and functional (Stump, 2001a, 2001b). Physically slope is represented as steepness which may be analyzed using algebra or trigonometry as well as viewed graphically using geometry. Assessing slope as a rate of change is referred to as functional. Both will require an understanding of rate or ratio and proportional reasoning. Teachers often refer to the physical aspect of slope such as the steepness of a ladder, road, ski slopes or building ramps (Stump,2001a, 2001b) using pictures, graphs, models and word problems.
Interviews, group discussions and questionnaires were
completed by part or all of the participants in the study that included 28
calculus students, 28 pre-service secondary mathematics student teachers, 50
in-service mathematics teachers and 14 mathematics educators (Zaslavsky, et al.,
2002). Zaslavsky, et al.
categorized slope into two different areas, analytic and visual. Their study
focused on the difference between slope of a line and slope of a function.
Analytically slope does not change when represented on a coordinate where the
independent and dependent axis are scaled differently, a non-homogeneous scale.
Correlating with Stump’s (2001a, 2001b) view, slope may be computed
algebraically using the difference quotient, using limits and calculus or
recognized as the coefficient of the variable x in an equation written in the
slope-intercept form. The rate of change view is placed under analytic by
Zaslavsky et al. which corresponds to Stump’s (2001a, 2001b) functional view
point. Visually slope is a property of a graph therefore the scale used when
graphing will make a difference in the interpretation. Slope as a visual
interpretation is determined using the angle
,
located between the x-axis and the given line, and the trigonometric tangent
function or the rise over run concept. Both visual interpretations by Zaslavsky
et al. correlate with Stump’s (2001a, 2001b) view of slope however the latter
did not mention the importance of scale. Visual slope (slope of a line) and
analytic slope (rate of change of a function) are not linked in any significant
and consistent manner (Zaslavsky et al.).
Rise
over run, difference of y values divided by the difference in x-values and as
the coefficient of the x term in the slope-intercept form of an equation are the
common references made to slope (Walter & Gerson, 2007). These approaches do
not aid students in transforming their understanding of slope as a rate of
change or to the positioning of a line on a graph (Walter & Gerson). Walter
and Gerson investigated slope by building on previous knowledge with a group of
six elementary school teachers, experience ranged from five to twenty-five
years. The participants used Cuisenaire Rods to represent a rate of change using
repeated addition. Extending their physical representation, the teachers created
a table of values and later constructed a graph. Teacher discussions included
the physical and functional concepts also addressed by Stump (2001a, 2001b) and
the visual and analytical ideas by Zaslavsky et al. (2002). Scale factors
utilized by the teachers using the Cuisenaire Rods provided rich conversations
about the relationship between physical or real-world situations and their
representations (Walter & Gerson) which is similar to the concern raised by
Zaslavsky et al. when employing non-homogeneous illustrations.
Rate of change, the functional aspect discussed by Stump
(2001a, 2001b), is the focus of an article by Farenga and Ness (2005) as they
investigate and make connections with slope. Farenga and Ness state that
constant, average, and instantaneous are three types of rate of change with each
the ability to be demonstrated visually and algebraically. Along with the
various types of rates of change came the importance of understanding the
concept of rate. The comparison of two different quantities, rate, are numerous
and have many applications in science as well as mathematics. Misunderstandings
surface with the many varied approaches including traditional and constructivist
methods.
Slope Misunderstandings
Procedural concerns
about formulas.
Teachers are generally more focused on the procedural and computational aspects of slope (Noh, 2004; Zaslavsky et al. 2002). This lack of conceptual knowledge is revealed by Stump (2001b), Simon and Blume (1994) and Walter and Gerson (2007) as they discuss the formula, why and how it is used and the relationship of slope to the equation of a line. Students involved in the study conducted by Stump (2001b) were presented with a model of a ramp and asked to determine slope. Eight of the 22 pre-calculus students stated various formulas not connected to slope. One student wanted to find the length of the hypotenuse when a right triangle was drawn while another student gave the formula for the area of a triangle. During another task students were given the slopes of three ramps and asked to describe the differences. Nineteen of the students did use the words slope, steepness and angle however some described the slope as a relationship between the height and length or just compared the heights of the three ramps.
The idea of steepness being determined by height and length was also part of a study conducted by Simon and Blume (1994) with 26 prospective elementary teachers. The teachers were given a model of a ski slope to determine the slope by investigating two methods. One of the methods involved the ratio of height to length, the ratio method, while the second method involved the difference in the two values, the difference method. A conceptual understanding of slope was developed through the comparison of the two approaches that lead to a better understanding of the formula. Stump (2001b) posited that even though students had been exposed to formal instruction, the conceptual understanding of slope had not been constructed. Incorrect usage of the formula or deciding on the correct formula to implement was one of the results of the student’s lack of understanding.
The six elementary school teachers involved in the study by Walter and Gerson (2007) created a table of values for y = 3x and y = 3x + 5. They noticed that a value of three had been added to each subsequent dependent value but the misunderstanding arose because the equation revealed that three was multiplied by the independent variable instead of being the constant value in the equation. Slope was seen as the constant difference or pattern instead of as a proportion. Teachers reasoned through the discrepancy by developing a recursive formula with aid from the instructor. This was done in steps by first using Cuisenaire Rods to model equivalent and non-equivalent fractional values. This enabled the participants to discuss steepness using concrete manipulatives which were then transformed into abstract symbolization. Inconsistencies arose as one of the participants referred to slope as mx + b instead of just the m. Physical models, activities or applications can aid in understanding slope but may also contribute to misunderstandings.
Relating slope to
physical models.
Stacking cubes, Cuisenaire Rods, model ski slopes and bicycle wheels may all be used to determine slope or rate of change. Physical models bring other concerns to the forefront. Human error in the building or creating of the model, human error when timing movement, comprehension of the activity that might lead to misinterpretation of what to model and even if the materials used in models have the same characteristics. The Cuisenaire Rods employed in the study by Walter and Gerson (2007) created some misunderstandings that the participants needed to rectify before proceeding on their study of slope. Rods chosen to represent a unit were different when the participant compared a slope of one-half to the slope of two-thirds. Comparisons were made between the various colored rods that eventually led to graphing on a homogenic plane. Gregg (2002) found that many university students understand rise over run and recognized that slope was the coefficient to the x in an equation but lacked the knowledge or experience to explain the concept or make relationships. This led to an activity involving stacking cubes in towers by stages and discussing the idea of slope similar to Walter and Gerson (2007).
Gregg’s (2002) university students immediately recognized a pattern found using the stage/position of the tower and how many cubes it took to create the tower. Through questioning students were led to determine the constant differences between the towers and the number of cubes it would take to build the zeroth tower. The position of the zeroth tower was illustrated as the position or floor it represented in the tower. Using their new knowledge students would attempt to predict additional values at higher stages. The stacking cubes activity assisted students in creating a meaning to linear relationships while connecting the graphical and the symbolic representations of these relationships (Gregg). Misunderstandings occurred when transfer was made from the concrete to the abstract which must be carefully planned by the facilitator. Gregg employed concrete objects that students and teachers could manipulate to aid in creating conceptual understanding whereas Simon and Blume (1994) used models that represented slope.
Prospective teachers investigated the height and length of a ski slope by exploring the ratio method and difference methods leading to an understanding of slope (Simon & Blume, 1994). The project revealed several misunderstandings of slope that included what is meant by positive and negative slope, which numbers to compare (length, width, height) and which of the methods work. Teachers determined several values for length and height using the difference method so that the differences were the same for each model. The group then determined if the structures created with the same difference value looked identical. It was also determined that for specific heights and lengths that a negative value might be the result. The misunderstanding of slope as the difference between the values was compounded by the confusion of what the negative result meant. A discussion about negative slopes was pursued with teachers commenting that it had no meaning, that one would need to ski on the underside of the slope or that the slope went underground. Teachers dismissed the difference method since it resulted in many conflicting ideas. The ratio method was investigated with the teachers creating many slope models having the same ratio of height to length.
Simon and Blume (1994) revealed that the teachers verified that the ratio method produced ski slope models with the same steepness. The slopes resulting from the same ratio confirmed the concept of invariance and the understanding that an isomorphic relationship must exist between the model and the mathematics. Models built and/or drawn helped the teachers understand slope (Simon & Blume) however using physical real world objects such as a bicycle wheel (Stump, 2000) has also helped students to become involved in their own learning.
Pre-Calculus students recorded the number of bicycle wheel rotations and pedal rotations for a specified length of time for various gears (Stump, 2000). Scatter plots were created for each of the gears along with the determination of the line of best fit. Of the twenty-two students involved in the activity only eight were able to make relationships between the wheel rotations and pedal rotations. The remaining students comments were very general however when asked specifically about the rate of change or slope only one student gave a numerical answer.
The task revealed a gap in student understanding of rate of change and that time needs to be spent in the classroom making connections between multiple representations and situations involving rates (Stump, 2001b). Asking thought provoking questions increases communication and encourages students to make relationships (Stump). In a global society students need to be able to apply their learning to new situations (NCTM, 2000). Having the opportunity to control their own learning, monitor progress and define goals encourages students to be more successful in mathematics (NCTM). Combining procedural competence with factual knowledge and conceptual understanding aids student’s ability to make connections between mathematical content and real world situations thereby becoming more self-sufficient.
Multiple Representations and
Applications
Gregg (2002) used stacking cubes, Walter and Gerson (2007) used Cuisenaire Rods and Simon and Blume (1994) used model ski slopes to illustrate the concept of slope. All of the activities challenged the teachers as they created the representations and wrestled with the isomorphism between the representation and the mathematics. In addition to the above physical applications, slope can be represented using multiple activities that include physical movements, objects, and stories or real life situations.
Printz (2006) employed multiple representations for slope that includes body modeling, using a model buggy and analyzing a table of values and graphs. She posited that “Students need the benefit of several hands-on activities and the benefit of multiple representations (graphs, equations, tables) in order to grasp the meaning of abstract ideas…” (p. 262). Representations of a concept include symbolic, graphical and tabular but also go further to include various types of real world situations that model the mathematics being learned. Printz had middle school students investigate the movement of a battery operated buggy that traveled at a constant velocity which led into body modeling, physical movements that illustrate a graph. Students were encouraged to discuss their findings and justify their statements. Whether it was the buggy or body modeling, tables and graphs were produced so that connections could be made between all by asking and analyzing where was the beginning, which direction did it move and how fast was the movement. Printz approached slope from a rate of change outlook while steepness was the approach taken by Andersen and Nelson (1994).
Physical objects such as ramps, stairs and sidewalks were measured using meter sticks with a focus on the horizontal and vertical measurements, what each stood for and the unit of measure (Andersen & Nelson, 1994). Several measurements were taken with each of the objects and then transferred to a geoboard to illustrate the steepness. Small group work extended to students understanding the slope of a horizontal and vertical line and why the slope was either zero or undefined. Misunderstanding could occur due to how students measured each of the objects or how they wrote their answers, as a ratio, a decimal or a percent. Another concern is the vocabulary that Andersen and Nelson used when discussing the activity with students. The word “steepness” was used when referring to a positive slope and only when a negative situation occurred did the vocabulary switch to “slope.”
Unique learning activities that encourage student participation are given by Johnsen and Wilkerson (2003). The creation of a unit of study on slope involved students and developed conceptual understanding at the same time (Johnsen & Wilkerson). From reading and illustrating Aesop’s fable “The Tortoise and the Hare” to researching roller coasters and creating mock graphs to filling up a bathtub full of water or drawing the coordinate plane on the floor are all activities that were chosen for the unit (Johnsen & Wilkerson). Students, after reading the fable, presented it as a newscast so that all understood the progression of activities. This was followed by having the students recreate the progression of events pictorially, discussing the rate of change of each of the stages in the race. The second lesson involved researching roller coasters, creating a mock roller coaster out of rows of chairs and then simulating the ride. Taping of the lesson allows students to relate time to the action of the roller coaster and to later create a graph of the simulation. During trial simulations prior to taping, students get to tell the conductor how they would like to see the ride change so that there is a steeper drop or a slower incline, enhancing conceptual understanding. Filling a tub full of water and letting some out is the third lesson which involves time and height of water. Students are required to write about the rate of change of the water with regard to the time.
Another physical interaction with slope similar to body modeling (Printz, 2006) is walking the “steps” that create slope (Johnsen & Wilkerson, 2003). The re-creation of a coordinate plane on the floor is a transition from physically modeling slope to the formula (Johnsen & Wilkerson). Two students hold an end of a rope and stand on a “point”. A third student must walk the rise and run while a fourth student records the position data. Transferring the data to a graph along with the lengths of the rise and run to touch the line assist students in making the connections. Additional lessons are discussed by Johnsen and Wilkerson that lead to writing equations however prior to their assessment students used technology to model positive and negative slopes.
Technological Applications
Technological applications include computer applications and software and handheld calculator device and its’ applications. Computer applications include Microcomputer-Based Laboratory (MBL) (Lapp & Cyrus, 2000; Wilhelm & Confrey, 2003). Calculator applications include the Calculator-Based Ranger (CBR) and/or the Calculator-Based Laboratory (CBL) (Johnsen & Wilkerson, 2003; Kwon, 2002; Lapp & Cyrus, 2000). Computer software include spreadsheets and the program Bank Account (Wilhelm & Confrey, 2003; Horton, 2000) while the graphing calculator device might be implemented independently or with the CBR or CBL (Demana & Waits, 1988; Zaslavsky et al., 2003).
The unit on slope created by Johnsen and Wilkerson (2003) concluded with a lesson implementing data collection devices such as the CBR and the CBL. Probes connected to the CBL allow data such as temperature and pressure to be recorded on a graphing calculator. CBR’s are motion detectors that connect to the graphing calculator. Data is collected as a student walks toward or away from the unit and stored in the calculator for future use. Visual representations of the data include graphical distance-time, velocity-time and acceleration-time displays. Tables are also produced revealing all data. Students had already experienced many activities where they had modeled scenarios involving movement so Johnsen and Wilkerson’s concluding activity requested the student to walk at a constant rate creating a linear graph. This involved students acknowledging the movements or speed needed to create a line that would stretch across the screen. Lapp and Cyrus (2000) also implemented the CBR and CBL but added an additional device, the Microcomputer-Based Laboratory (MBL).
The CBR, CBL and the MBL assist students in relating real world situations to mathematics and science (Lapp & Cyrus, 2000). Brasell (as cited by Lapp & Cyrus) states that real time graphing of data aids in student understanding of the concept and that even a small delay between the physical action and the produced graph will reduce understanding. Once students have analyzed graphs produced in real time they are better able to transfer their knowledge to other situations (Lapp & Cyrus). The ability to jump between a representation, graph or table, and the physical action is a vital skill in science (Lapp & Cyrus). Using technology encourages and enables students to interpret actions almost immediately according to Lapp and Cyrus. Beichner (1990) went one more step and recorded students walking so that the video could be viewed simultaneously with the graphical interpretation to aid in analyzing the movements for rate of change interpretations.
An MBL is more cumbersome to use as a motion detector or with
the probes (Lapp & Cyrus, 2000) however the application does create real
time graphs and saves information for future reference allowing student’s time
to make predictions and explain the visual. Prior to using the technology high
school students were given time-distance graphs and asked to illustrate each by
walking (Lapp & Cyrus). Many students were confused with the graphs and
walked by matching their movements to the shape on the graph instead of walking
back and forth in a straight line (Lapp & Cyrus). A data collection device
was then introduced to students. Interacting with the device increased student’s
understanding of rate of change by enabling the student to relate his or her
movements to the graphical interpretation (Lapp & Cyrus). An example would
be to use a CBR and the distance-time application that is built into the unit.
The calculator reveals a graph that a student will attempt to simulate using the
CBR in real time mode. The students walking distance graph is produced on the
same coordinate plane as the original graph thereby allowing the student to
compare his or her movements in the display. Data collection devices are not an
answer to learning conceptual knowledge about slope; it is an aid (Lapp
& Cyrus).
SimCalc,
software that can be implemented using Texas Instruments graphing calculators or
computer applications like JAVA, allows the user to not only record and view
data but to interact with the resulting graph (Moreno-Armella, Hegedus & Kaput, 2008). This
interaction encourages ownership in the student’s learning and becomes
personally meaningful. Moreno-Armella et al. states that the ability to interact
with the depiction of collected data is a form of representation they termed as
an “executable representation” (p. 106) versus static, traditional
representations such as manually created graphs. The traditional forms do not
encourage communication of mathematics among students. Social relationships are
important to student’s therefore encouraging communication of mathematical
representation manipulations, conjectures, and predictions will increase student
understanding (Moreno-Armella et al.). Working together to record, discover, and
analyze data or movements gives students’ confidence as they progress toward a
common goal in the learning journey.
Kwon (2002) conducted a study of 590 students in Korea using CBR’s as an aid to learning slope conceptually. The study group consisted of 428 middle school students taking Algebra I and a control group of 162 Calculus students. All students were pre-tested to determine their knowledge and understanding of graphs and slope with results revealing Calculus student’s knowledge exceeded Algebra I students. Labs were conducted using the CBR with Algebra I students and then all were again assessed. The final assessment revealed that Algebra I student knowledge exceeded Calculus students. Kwon’s study showed that the use of technology to aid in the visualization of slope and rate of change cleared misunderstandings of the concept. Wilhelm and Confrey (2003) also implemented computer based motion detectors along with a computer program.
Algebra I students in an inner city school in Texas were
taught slope as a part of a unit on the math of change (Wilhelm & Confrey,
2003). Three 90 minute class periods were spent using the motion detector
followed by two class periods using the computer program Bank
Account. Four
of the students were chosen to be interviewed regarding their experiences and
gained knowledge. Wilhelm and Confrey concluded that the students did have some
difficulties connecting rate of change to the accumulation concept presented in
the computer program with only one of the students fully understanding both. A
deep understanding of rate of change and accumulation was recognized in all but
one of the students. “Technologies used to assist in rate of change
understanding need to be contextualized in multiple areas, not just motion”
(Wilhelm & Confrey, p. 904), since working in a single context inhibits
students fully understanding a concept. Multiple contexts and multiple
technologies will enhance learning and steer students and teachers away from
procedural lessons (Wilhelm & Confrey). Instead of implementing multiple
technologies Demana and Waits (1988) and Zaslavsky et al. (2002) employed only
graphing technology.
Graphing technology allows students to zoom in and out on the screen in order to investigate the behavior of a function (Zaslavsky et al., 2002). Students, teachers (pre-service and in-service), and mathematics educators involved in the study conducted by Zaslavsky et al. analyzed a line graphed on a homogeneous coordinate system and on a non-homogeneous coordinate system. Questions were asked about the slope of the function f, if the line was an angle bisector of the first quadrant and if they could determine the tangent of the angle between the line and the x-axis. All of the questions were followed up with requests for justification of their responses and each assessed according to the analytic and visual perspectives discussed previously. Many of the participants were confused as they knew the slope of the line graphed (y = x) and always assumed that the change in the scale factor supposedly would not affect the behavior of the function (Zaslavsky et al.).
Zaslavsky et al. (2002) surmised that the slope in a homogeneous plane was one whereas the slope in a non-homogeneous plane was one-third when the units were multiples of three on the y-axis and remained units of one on the x-axis. Making the unit values equivalent on the non-homogeneous plane without changing the line allows the algebraic and trigonometric computations to be assessed via the visual perspective (Zaslavsky et al.). An assumption is always made that a homogeneous plane is the standard when visualizing graphs and violated only for display purposes (Zaslavsky et al.). Comparing the same line graphed on homogeneous and non-homogeneous planes may be inappropriate for high school students but a good topic for discussion among in-service and pre-service teachers (Zaslavsky et al.).
Computer graphing technology can change the way students approach mathematics and help make challenging problems possible (Demana & Waits, 1988). With the good aspects come the concerns. Understanding what is seen on the screen and, more importantly, what is not on the screen can mislead students (Demana & Waits). Even though Demana and Waits do not specifically discuss slope in their article they do stress the importance of knowing function behaviors. Graphing programs may connect the last point plotted on one side of a vertical asymptote to the first point plotted on the other side of the asymptote leading to the display of a false line (Demana & Waits). End behaviors may also be misunderstood due to the window or scale size (Demana & Waits). “By analyzing numerous graphs of functions in a short period of time, students are able to build strong intuition and understanding about functions (Demana & Waits, p. 183).
Connections may also be made using a spreadsheet. Horton (2000) employed scenarios similar to Johnsen and Wilkerson (2003) with the exception that the values were recorded on spreadsheets. This enabled the students to manipulate data using formulas and analyze the results easier. Involvement in the activity resulted in secondary students building recursive formulas, applying repeated addition, as well as developing explicit formulas based on multiplication (Horton). Linking the repeated value with the factor and the graph and table enabled students to better understand slope.
Slope Represented as a Fraction, Decimal or
Percent
Andersen and Nelson (1994) were not the only authors to discuss writing and interpreting slope as a ratio, a decimal or a percent. The percent of grade on a highway was posed to teachers and high school students (Simon & Blume, 1994; Stump, 2001b); examples and non-examples of constant slopes were expressed with the various forms and then analyzed (Crawford & Scott, 2000). Confusion arose from going “backward” from the given slope or rate of change to describing or interpreting the situation.
Simon and Blume (1994) worked with the prospective elementary school teachers to conceptualize slope using the difference and ratio method. While exploring the concept with models teachers were questioned about the meaning of a 7% grade and would drivers be able to understand its’ meaning. Many considered the percentage as a rating and not as a measure of steepness or a ratio of two quantities. The teacher’s lack of conceptual understanding of ratio or proportion was acknowledged as they lacked the ability to make connections between the mathematics and real world situations. Only three of twenty-two students explained using the term ratio while others suggested that it might be related to a circle (Stump, 2001b). Stump also asked students to match decimal approximations of slope to its model. Half of the students did so correctly while only six could explain what the numbers meant or referred to in the given situation.
Crawford and Scott (2000) used the alternate forms of expressing slope with examples and non-examples of constant rates of change to illustrate slope. Their approaches illustrate what Walter and Gerson’s (2007) study of elementary school teachers did with tables, exploring addition and multiplication of dependent and independent values. Algebra I students was given a scenario where a pair of $30 jeans would increase in cost at 4% per year (Crawford & Scott). Students created tables and graphs to illustrate the data acknowledging that the rate of change between data points did not remain constant. Comparing the non-example with the cost for a company to buy and then rent a canoe to customers aided in creating the conceptual understanding of slope.
Teacher Issues
A shift from the procedural to applications will aid in students conceptual understanding of the topics (Crawford & Scott, 2000). Before teachers can teach conceptually they must understand and build the mathematics they are expected to teach (Simon & Blume, 1994; Thompson & Thompson, 1996). This will not happen with teacher preparation programs including more higher level mathematics courses. Prospective teachers need to be taught using methods that will be implemented to teach their future students (Simon & Blume).
Conceptually approaching slope, Crawford and Scott (2000) suggest visualizing the situation first and then creating graphs and tables to represent the situation. Students need to verbalize the situation, the graph and the table and finally represent the situation with symbols. Multiple representations are not enough (Swafford & Langrall, 2000). Connections need to be made between the multiple representations and to the context of the problem in order for students to understand the concept. The concern in doing so is the difficulty students have in finding patterns that are algebraically useful. Tables, the least used representation, may help in discovering the patterns, interpreting situations and making sense of the given problem. Further difficulties arise when searching for good problems that make connections and encourage experiential learning (Johnsen & Wilkerson, 2003). A hands-on approach to learning mathematics encourages student involvement and enables each participant to take ownership in his or her own learning (Johnsen & Wilkerson). Students ask higher level questions and teachers become facilitators in the learning process.
Future Studies
Studies gathered have focused on some students, Algebra I through Calculus; however most participants have been teachers or prospective teachers. Educating pre-service teachers through their course work and in-service teachers through professional development opportunities followed by extended guidance and assistance will eventually reach and affect more students. Walter and Gerson (2007) posit that in-service professional development sessions should center on teachers as the driving force, collaborating on mathematical ideas and habits. Content is important but needs to be sustained through reflection of what was intended and exactly what was achieved (Thompson & Thompson, 1996). Due to the complexity of determining or creating problems that are hands-on and encourage students to become involved in their learning, teachers needs to collaborate, research and provide peer support. Teacher educational programs and professional development sessions can plant the seed for experiential learning but many times the work expectations of pre-service and in-service teachers is not focused on multiple representations or presenting slope as a rate of change and steepness (Stump, 2001a), hence the need for the follow up guidance and assistance. Changing the way one thinks does not change with one example or application. Longitudinal studies that follow teachers for several years to observe growth, obstacles and how they develop interactive lessons would be helpful to many teachers. Long range professional development sessions that will lead teachers through the process, providing support, need to be developed and assessed for teacher and student attitudes and student academic achievement. Lesson study could also be a component of professional development so teachers collaborate to create the best lesson and then facilitate the lesson with a focus on student response and interactions.
Moreno-Armella et al. (2008) also brought to the forefront the importance of social interactions among students. Research may include how students work together, mathematical communication and how it may or may not increase understanding. Student to student interactions need to be studied along with teacher-student interactions. Educating teachers on how to effectively manage interactive situations so that mathematics may be learned and understood should be considered for research for best practices.
Closing
Teachers instruct or facilitate learning mathematics by employing methods that reveal their own understanding (Thompson & Thompson, 1996). In order for students to understand slope, as well as other topics, conceptually, teachers need to delve deeper into the content and how student perceive concepts. Slope is more than a formula or the steepness of the line created that represents a situation. Teacher understanding of slope using the physical and functional or the analytic and visual categories described (Farenga & Ness, 2005; Stump, 2001a, 2001b; Walter & Gerson, 2007; Zaslavsky, et al., 2002) increases his or her ability to make connections and meet student needs. Along with increasing content knowledge comes an awareness of some misunderstandings of the slope concept that may arise while teaching and exploring slope. Encouraging communication, math talk, among students allows the teacher to acknowledge how a student is thinking and processing the knowledge. Teachers become listeners instead of the teller.
References
Andersen, E. D. & Nelson, J.
(1994, January). An introduction to the concept of slope. The
Mathematics Teacher, 87(1), 27.
Beichner, R. J. (1990). The effect of simultaneous motion presentation and graph generation in a
kinematics lab. Journal of Research in Science Teaching, 27(8), 803-815.
Berg, C. A. & Smith, P. (1994). Assessing students’ abilities to construct and interpret line
graphs: Disparities between multiple-choice and free-response instruments [Abstract]. Science Education, 78(6), 527-554.
Crawford, A. R. & Scott, W. E.
(2000, February). Making sense of slope. The Mathematics
Teacher, 93(2), 114-118.
Demana, F. & Waits, B. K. (1988, March). Pitfalls in graphical computation, or why a single
graph isn’t enough. The College Mathematics Journal, 19(2), 177-183.
Farenga, S. J., & Ness, D. (2005, April/May). Science and algebraic thinking part 1: Rates and
change. Science Scope, 28(7), 58-61.
Gregg, D. U. (2002, May). Building students’ sense of linear relationships by stacking cubes.
The Mathematics Teacher, 95(5), 330.
Horton, B. (2000, May). Making
connections between sequences and mathematical models. The
Mathematics Teacher, 93(5), 434.
Johnsen, C. & Wilkerson, T. L. (2003, October). My journey toward a new slant on slope. The
Mathematics Teacher, 96(7), 504.
Kwon, O. N. (2002, February). The effect of calculator-based ranger activities on students’
graphing ability. School Science and Mathematics, 102(2),
57-67.
Lapp, D. A. & Cyrus, V. F. (2000, September). Using data-collection devices to enhance
students’ understanding. The Mathematics Teacher, 93(6), 504.
Moreno-Armella, L., Hegedus, S., & Kaput, J. (2008). From static to dynamic mathematics:
Historical and representational perspectives. Educational Studies in Mathematics, 68(2), 99-111.
National Council of Teachers of
Mathematics. (2000). Principles and
standards for school
Mathematics. Reston, VA: NCTM.
Noh, J. (2004). “Understanding rate
of change: Stories of secondary teachers.” Paper presented
at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <PDF>. Retrieved November 2008 from http://www.allacademic.com/meta/p117575_index.html
Printz, J. (2006, May). The buggy lab: Comparing displacement and time to derive constant
velocity. School Science and Mathematics, 106(5), 261-266.
Simon, M. A. & Blume, G. W. (1994). Mathematical modeling as a component of understanding
ratio-as-measure: A study of prospective elementary teachers. Journal of Mathematical Behavior, 13, 183-197.
Stump, S. L. (2000, December). Doing
mathematics with bicycle gear ratios. The
Mathematics
Teacher, 93(9), 762.
Stump, S. L. (2001a). Developing preservice teachers’ pedagogical content knowledge of slope.
Journal of Mathematical Behavior, 20(2), 207-227.
Stump, S. L. (2001b, February). High school precalculus students’ understanding of slope as
measure. School Science and Mathematics, 101(2), 81-89.
Swafford, J. O. & Langrall, C. W. (2000, January). Grade 6 students’ preinstructional use of
equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112.
Thompson, A. G. & Thompson, P. W. (1996, January). Talking about rates conceptually, Part II:
Mathematical knowledge for teaching. Journal for Research in Mathematics Education, 27, 2-24.
Walter, J. G. & Gerson, H. (2007). Teachers’ personal agency: Making sense of slope through
additive structures. Educational Studies in Mathematics, 65, 203-233.
Wilhelm, J. A. & Comfrey, J. (2003). Projecting rate of change in the context of motion onto the
context of money. International Journal of Mathematical
Education in Science and
Technology, 34(6), 887-904.
Zaslavsky, O., Sela, H., & Leron, U. (2002, January). Being sloppy about slope: The effect of
changing the scale. Educational Studies in Mathematics, 49(1), 119-140. Retrieved
January 2, 2009, from Academic Search Complete database.