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In a World of Data, Statistics Counts
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by Richard Scheaffer
University of Florida
Gainesville, Florida
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| | Statistical Revolution Versus Statistical Literacy
Science entered the nineteenth century with a firm philosophical vision that has been called the clockwork universe. By the end of the nineteenth century... the clockwork universe lay in shambles. Gradually, science began to work with a new paradigm, the statistical model of reality. By the end of the twentieth century, almost all of science had shifted to using statistical models.
Popular culture has failed to keep up with this scientific revolution.
-- David Salsburg, The Lady Tasting Tea
On the one hand, nearly all of science views the world from a statistical perspective; on the other, most citizens of the land do not. "Despite years of study and life experiences in an environment immersed in data, many educated adults remain functionally innumerate." So say the authors of a recent publication entitled Mathematics and Democracy: the Case for Quantitative Literacy. The report from a cross section of educators goes on to say that "most U.S. students leave high school with quantitative skills far below what they need to live well in today's society. Quantitative Literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. These are the skills required to thrive in the modern world" (National Council on Education and the Disciplines 2001). Forerunners to this report speak of the same needs among adults of this country. For example, the chief statistician of the U.S. Office of Management and Budget, Katherine Wallman, stated in 1999, "Our citizens encounter statistics at every turn in their daily lives. Yet often they are unequipped with the statistical literacy required to evaluate the information presented to them."
The need for improved quantitative and statistical literacy has not gone unnoticed among statistics teachers and educational leaders across the country. At the school level, the Principles and Standards for School Mathematics produced by the National Council of Teachers of Mathematics (NCTM) in 1989 and revised in 2000 contains a data analysis and probability theme that runs throughout grades K to12. The motivation for including such a theme is partially explained as follows: "The amount of data available to help make decisions in business, politics, research, and everyday life is staggering. Students need to know about data analysis and related aspects of probability in order to reason statistically -- skills necessary to become informed citizens and intelligent consumers." A similar theme is echoed by the developers of the National Assessment of Educational Progress (NAEP), who recommend that 25 percent of the high school exam be devoted to data analysis and probability in 2004. These are but a few of many movements that are attempting to nudge the school mathematics curriculum toward content that includes deep and modern concepts of statistical literacy. At the college level, enrollments in introductory statistics courses in statistics and mathematical sciences departments saw a 54 percent increase over the decade of the 1990s, according to a recent report from the Conference Board of the Mathematical Sciences (2001). (In comparison, mainstream Calculus I enrollments remained flat over the decade.) The increase in statistics enrollments is due not only to the increased importance of the subject within many disciplines but also to the serious efforts being made to change the pedagogical climate of Stat 101 so that it moves from dull and dreaded to lively and loved. A recent survey by Joan Garfield (2002) reveals a strong movement toward reforming introductory statistics courses. She reports, "Reform efforts [most commonly, use of technology, active learning, and alternative assessment methods] appear to be affecting many introductory statistic sources.... Most faculty members reported positive outcomes regarding changes made: more student satisfaction, more student learning, increased faculty enjoyment, and more sharing of ideas and methods with colleagues." The ingredients are present to effect a cultural change in statistical literacy, but a coordinated path to achieving this goal has not yet been articulated. The AP Program can help.
AP Statistics at the Juncture
AP Statistics sits at the juncture of the school and college curriculum. Its success has its roots in the quantitative literacy program developed by the American Statistical Association (ASA) in the 1980s, which had considerable influence on the statistics (actually, data analysis and probability) strand of the NCTM Principles and Standards. These efforts, focused by the desire of the College Board to add another mathematical science course to the AP Program, paved the way for AP Statistics to get off to a flying start because of the interest already generated among teachers and students. (See the 1999 article by Scheaffer, Roberts, and Watkins.) The modern content and active learning style that permeates the AP Statistics program has, in turn, challenged some colleges to revamp their introductory statistics courses and to consider adding other courses in modern statistics for undergraduates who have the need, interest, and skill to go beyond the first course. (See the July 2002 Journal of Statistics Education below in "See also" for a review of current activities in improving undergraduate statistics education.) Now that the AP Statistics program is firmly established, it can (and should) return a favor by influencing the direction and strength of the K-12 strand in data analysis and probability.
The goal of the K-12 strand in data analysis promulgated by NCTM and others is to provide students with basic education in quantitative literacy, or numeracy, and statistical thinking that will help them function successfully in a world of data. An added benefit is that this statistical approach also provides motivation and illustration for many other topics in the mathematics curriculum. Building on that background, the goal of the AP Statistics program is to go beyond basic numeracy and provide a modern college-level course for high school students who are ready for the challenge. Such a course should provide students with the background necessary to succeed in the next level of statistics courses (or courses requiring statistical knowledge) offered by the college programs they enter. Colleges and universities do need to be educated about the AP Statistics program, however, because the program is relatively new, and the college course for which a student might receive credit or placement may be in any one of several departments. In short, one can view the idealized K-16 curriculum as a system in which the teachers of the data analysis strand in the schools can look ahead to the AP course to see where they should be heading, the college instructors can look to the AP course to see how they should be expanding, and the AP Statistics teachers can influence both groups by paying attention to the skills students have (or do not have) coming into their courses and following up on their graduates as they enter colleges and universities.
This may put an unfair burden on the AP teachers, but professional societies (NCTM, ASA, MAA, and others) can help by working together to provide resources for improving statistics education through various forums for the exchange of information, such as sessions at meetings, special topic workshops, and exciting Web pages. An expanded Center for Statistics Education at ASA is one possibility for opening the dialogue and building consensus on guidelines for a K-16 statistics curriculum. The National Science Foundation offers numerous programs that could provide support for innovative programs in developing a connected statistics curriculum.
Teacher Education As the Key
The key to improved teaching, especially of topics relatively new to the curriculum such as data analysis, is improved skill of the teacher. The importance of this topic to all of the mathematical sciences is discussed rather thoroughly in the 2001 report entitled The Mathematical Education of Teachers (MET) from the Conference Board of the Mathematical Sciences (CBMS). The report calls upon mathematical sciences departments to take seriously their role in the education of future teachers (K-12). Statistics is singled out as an area that needs special attention, as indicated in the following quotes from the report: "Statistics is the study of data, and the daily display of data by the media notwithstanding, most elementary teachers have little or no experience in this vitally important field," and "At the high school level, traditional emphasis on probability-based statistical inference has given way to focusing on data analysis to gain insight into problematic situations." Modern data analysis was not part of the college preparation of most teachers in the schools today. Although there exist many curriculum development projects that provide excellent teaching materials and ideas on active learning built around real data, there appears to be little effort in making these part of the formal academic education of the teachers who will be called upon to teach the data analysis strand or the AP Statistics course. Perhaps the same network that solves the curriculum problem mentioned above can take on this challenge as well.
Data Analysis and Mathematics?
Why has statistics in the schools come of age as part of the mathematics curriculum? Why is it not in science or the social sciences? The simple answer is that the mathematics community was well organized to make changes to their curriculum as society advanced to the information age and was open to accepting statistics as part of their charge. (The AP Statistics program came about largely though the efforts of the AP Calculus Committee.) There are, however, some conflicts between mathematics and data analysis that can cause the latter to sit less than comfortably in the home of the former. Data analysis is about context; mathematics is about pattern and logic free of context. The data analyst must make a decision (inference) based on partial knowledge; a mathematician wants to prove (deduce) results based on a set of general principles. A data analyst is willing to make errors (the chance of which can often be measured), and the mathematician wants to reason without error.
But, the process of data analysis involves the use of mathematics (number concepts, geometry, algebra, and functions), and the teaching of data analysis can provide practical examples of the interrelationships among these various areas. Data are expressed in numbers, so it is clear that number concepts can be reinforced by having students collect and analyze data. Data and/or probability distributions often are expressed by plots in which area represents relative frequency or probability (envision a bar graph or histogram). In using spreadsheets to collect and store data for analysis, the distinction between "variable" and "case" becomes clear, and new variables are often represented in terms of old ones through algebraic expressions. At a higher level, possible relationships between variables might be modeled by linear (or more complicated) functions. Virtually all of the content areas of K-12 mathematics can be illustrated and motivated by data analysis, and in the process the students are likely to be captivated by the fact that the mathematics has practical application in a problem of interest to them. A few quotes from teachers experienced in infusing their mathematics classrooms with data may help make the point. These teachers were using modern data analysis materials developed under a National Science Foundation grant in the teaching of algebra; there is now a fairly large collection of such material available.
- "The students DO the work as presented in the... materials. I believe that the students do the work because it has meaning for them. For those students I taught in the math in the workplace class just doing anything in class was an accomplishment, but these students not only did what I asked them but they also tested well at the end of the units."
- "The students pay attention to the material... [U]sing data to figure out what a car costs is something high school students pay attention to. Teaching students to write an equation of a line and to discuss the meaning of the slope has never been this easy! Actually I'm not sure that students ever understood this concept well before."
- "The materials allow for the students to construct knowledge based on their experiences, and these materials provide activities and experiences to guide the students to good concept-based skills. The students understand what and why they are doing things." (Kathy Harris, Caldwell, Idaho)
- "Almost all of the students were amazed by the fact that some of the mathematical concepts that they study (logs and exponentials) are actually used in such data analysis situations. I must also say that I find it very exciting to engage in these topics as well!" (Mark Zilinskas, Indiana, Pennsylvania)
- "I really love the approach it takes of using real data and using a rate of change instead of a slope. This allows students to understand the graphs they are doing and actually be interested in the results. I intentionally used these materials with one class while using the old slope-intercept with another and found the results varied greatly in that the (data-oriented) class seemed to retain the information quite a bit longer. They also liked doing the work quite a bit more."
- "[One] module starts teaching probability as an extension of the relative frequency. Again, this approach helps the kids buy into the data which they simulate using coins, spinners, and skittles! It also lends itself towards a good understanding of the estimation involved when using probability and simulations." (Cathy Moore, Cascade, Idaho)
Even though statistics seems to have found a certain level of support within mathematics, no one objects to having more statistics in the science and social science curriculum as well. In fact, both statistics and mathematics would benefit from closer connections to all of the sciences, and the two fields should be mutually supportive of one another in building such bridges. In the spreading of statistics across the curriculum, care must be taken to keep roots in the statistics profession so that the modern flavor and vitality of the subject can be maintained.
Conclusion
Data are hot! Everyone -- students, teachers, parents, employers -- is interested in data, but few know how to collect and interpret data intelligently. Data is the basis of science, and statistical thinking is key to the scientific method, yet few graduates of high school and college understand how science works. For these and other reasons, statistics must become a major component of the modern K-12 mathematics curriculum and achieve a stronger presence in the undergraduate curriculum, as recommended by a wide variety of educational groups.
Since AP Statistics sits at the juncture of high school and college, its teachers and supporters can and should influence the curriculum in both directions so as to improve both the K-12 strand in data analysis and the college offerings in statistical science. Professional societies in the mathematical sciences should organize an arm that supports a cohesive, vertically integrated K-16 plan for high-quality statistics education, including an emphasis on improved education of future teachers.
Statistics is not the same as mathematics but uses mathematics as one of its main tools for practical problem solving. Nevertheless, at the school level, statistics is most readily seen as a part of the mathematics curriculum. Since statistics deals with applications and is one of the most widely used of the "mathematical sciences," it is relatively easily connected to many other components of the curriculum, K-16, especially the sciences and social sciences. Statistics teaching is easily adapted to active learning, much in the spirit recommended by modern cognitive science. All of this points to the fact that dividends can be reaped by the two disciplines -- statistics and mathematics -- working together and with other disciplines toward a horizontally integrated plan to increase the quantitative skills of future generations of students. It is this goal that must dominate the interest, energy, and resources of the statistics education community in the years ahead if the information age is to reach its full potential of informed decision making based on rational thought and quantitative evidence.
Richard Scheaffer is professor emeritus of statistics at the University of Florida. He was one of the developers of the Quantitative Literacy Project in the U.S. that formed the basis of the data analysis emphasis in the mathematics curriculum standards recommended by the National Council of Teachers of Mathematics. He directed the task force that developed the AP Statistics course and Exam, and served as its first Chief Faculty Consultant. Dr. Scheaffer is a Fellow and past president of the American Statistical Association, from which he has received a Founder's Award.
References
Conference Board of the Mathematical Sciences. 2001. Issues in Mathematics Education Volume 11: The Mathematical Education of Teachers. Providence: American Mathematical Society.
Garfield, Joan. 2002. Evaluating the Impact of Educational Reform in Statistics: A Survey of Introductory Statistics Courses. Final Report for NSF Grant REC-9732404.
National Council on Education and the Disciplines. 2001. Mathematics and Democracy: The Case for Quantitative Literacy. Princeton: Woodrow Wilson Foundation.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, Virginia: NCTM.
Salsburg, David. 2001. The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century. New York: Freeman.
Scheaffer, R. L., R. Roberts, and A. Watkins. 1999. "Advanced Placement Statistics -- Past, Present, and Future." The American Statistician 53: 307-320.
Wallman, Katherine K. 1999. "At the Intersection of Official Statistics and Public Policy: Confronting the Challenges," Amsterdam: Speech delivered at Celebrating the Centenary of the Netherlands Statistical Office.
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