What does the slope mean?

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Sep 2001 by 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. fgordon@iris.nyit.edu.

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 [1] 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.