What does the slope mean?
Gordon, Florence S
ADDRESS: Department of Mathematics, New York Institute of Technology, Old Westbury Campus – P O BOX 800, Old Westbury NY 11568-8000 USA. email@example.com.
ABSTRACT: The author describes some of the results of a comprehensive study comparing student performance in a reform/modeling version of college algebra/trigonometry to that in a traditional version of the course. The focus in this article is on student interpretation of the meaning of the slope of a line in a realistic context, where very different responses occurred depending on the kind of course the students were taking. The author then discusses the broader implications across all of mathematics education of the results of this study.
KEYWORDS: College algebra, college algebra/trigonometry, precalculus, modeling, slope of a line.
“What does the slope mean?” is a seemingly innocuous question. But, when one looks at students answers to this question, as you will see below, it is apparent that what we think they are learning can be radically different from what they actually learn.
The Mathematics Department at New York Institute of Technology (NYIT) recently conducted a multifaceted study comparing student performance in a reform version of college algebra/trigonometry based on mathematical modeling to that in a traditional college algebra/trigonometry course. Aspects of the study also included student attitudinal surveys, as well as performance, retention, and persistence studies in the follow-up Calculus I course for engineering and science students. In this article, we focus on just one small aspect of the study the student responses to a question on the meaning of slope and the significance of those responses in terms of their implications for all of mathematics education.
The complete report  by the external evaluator is available at
In the study at NYIT, two instructors taught sections of the traditional course while two others (including the present author) taught sections of the reform/modeling course. Enrollment in both of the reform/modeling sections and one of the traditional sections was about 20 students in each; the other traditional section had about a dozen students. The students in all sections were required to use graphing calculators. The traditional course had an algebraic drill-andskill format. The reform/modeling course emphasized conceptual understanding and real-world applications, with the algebra embedded entirely within the contexts of solving problems.
Students had no idea of the difference in the course sections when they registered, so that placement into the different classes was essentially random. However, the Mathematics Department does administer its own placement test, which is purely algebraic in nature. As part of the evaluators study, it was found that the students who registered for the two reform/modeling sections had considerably lower scores on the placement test (an average of 12.5 out of 20) than those who registered for the two traditional sections (13.6 out of 20). Thus, ironically, the students who took the traditional course with its emphasis on algebraic skill development actually had better algebraic skills to begin with.
The principal component of the college algebra/trigonometry portion of the study involved a comparison of student performance on 10 common questions on the two final exams. Because some faculty felt it would be unfair to the students in the traditional sections to ask any questions that emphasized conceptual understanding or substantial realistic applications, the common questions were, of necessity, essentially algebraic in nature, even though this had not been a major focus of the two reform/modeling sections. Moreover, some faculty wanted to be sure that the reform/modeling approach did not inflict undue “damage” to the students already weak algebraic abilities.
Surprisingly, it turned out that the students in the reform/modeling sections outperformed their peers from the traditional sections on seven of these 10 questions.
Only two of these common questions had any semblance of realistic content. One of them is in Figure 1. Out of a total of 10 points allotted for the problem in Figure 1, the students in the reform/modeling sections scored a mean of 9.14 with standard deviation of 1.38 while the students in the traditional sections scored a mean of 6.33 with standard deviation 3.71. A means/ANOVA t-test indicates that the two means were significantly different (F-ratio = 17.8202, ga-value = 0.0001).
Brookville College enrolled 2546 students in 1996 and 2702 students in 1998.
Assume that enrollment follows a linear growth pattern.
(a) Write a linear equation that gives the enrollment in terms of the year t (let t = 0 represent 1996).
(b) If the trend continues, what will the enrollment be in the year 2016?
(c) What is the slope of the line you found in part (a)?
(d) Explain, using an English sentence, the meaning of the slope here.
Incidentally, the entire problem on both final exams was worth a total of 10 points. The part asking for the interpretation of the slope was worth only 2 points. Thus, this portion by itself represents only a fraction of the difference in the overall scores (9.14 versus 6.33) between the two groups. In fact, it seems that most of those who had trouble interpreting the slope also had trouble using the equation of the line to answer the predictive questions posed. It is interesting to note that both groups had comparable ability to calculate the slope of a line. However, any graphing calculator and many commonly available software packages, such as Excel, can do that also. We believe that what should be more valuable to our students is the ability to understand what the slope means in context, whether that context arises in one of their other courses in mathematics, or courses in one of the quantitative disciplines, or eventually on the job.
More significantly, it is apparent from these results that, unless explicit attention is devoted to emphasizing the conceptual understanding of what the slope means, the majority of students are not able to create viable interpretations on their own. And, without that understanding, they are likely not able to apply the mathematics to realistic situations. In this context, many of us have repeatedly heard complaints from colleagues in other disciplines about students seemingly not having learned key mathematical ideas and techniques, often the equation of a line. But if the results of the present study are indicative of students in general, and there is no reason not to think so, then these complaints make sense. Too many mathematics courses stress the manipulative technique for finding the equation of a line without emphasizing the underlying conceptual understanding or realistic contexts in which such problems would arise in practice.
For that matter, in most of the other disciplines, linear functions do not arise in the form we usually teach: Find the equation of the line through the points (1, 3) and (5, 11). In all other disciplines, one typically faces a collection of data on some quantities of interest that follows a roughly linear pattern and one has to find and use the (regression) line that best fits the data. If students have as much difficulty just giving a meaning to the slope of a line as this study indicates, it is no wonder that they are unable to make the connection between what they learn about lines and linear functions in their mathematics classes and what they are doing in their other courses.
Moreover, if students are unable to make their own connections with a concept as simple as the slope of a line (which they have undoubtedly encountered in previous mathematics courses), it is unlikely that they will be able to create meaningful interpretations and connections on their own for more sophisticated mathematical concepts. For instance, what is the significance of the base (growth or decay factor) in an exponential function?
What is the meaning of the power in a power function? What do the parameters in a realistic sinusoidal model tell about the phenomenon being modeled? What is the significance of the factors of a polynomial? What is the significance of the derivative of a function? What is the significance of a definite integral? (Interested readers may want to pose such a question on one of their own tests.)
On the basis of this study, it is clear that we cannot simply concentrate on teaching the mathematical techniques that the students need. It is just as important to stress conceptual understanding and the meaning of the mathematics. This can and should be accomplished by using realistic, contextual examples and problems and by forcing the students to think, not just to manipulate symbols. If we fail to do this, we are not adequately preparing our students for successive mathematics courses, for courses in other disciplines, and for using mathematics on the job and throughout their lives.
The work described in this article was supported by the Division of Undergraduate Education of the National Science Foundation under grant #DUE9555401 for the Long Island Consortium for Interconnected Learning. However, the views expressed are not necessarily those of either the Foundation or the project.
1. Colley, Kabba. 2000. Comparing College Students Performance in Traditional and Reform Pre-Calculus and Calculus Courses: An Evaluation Report. (http://iris. nyit. edu/math/eval.doc.)
Florence S. Gordon is professor of mathematics at New York Institute of Technology. She received her PhD from McGill University. She was a member of the working group on the Math Modeling/PreCalculus Reform Project and is a co-author of the project text, Functioning in the Real World: A PreCalculus Experience. She is the co-editor of the MAA Notes volume, Statistics for the Twenty First Century, the co-author of Contemporary Statistics: A Computer Approach, and is the author of over 40 articles on statistical research and mathematical and statistical education. She is also a local-project director of the NSF-supported Long Island Consortium for Interconnected Learning in Quantitative Disciplines.
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