STATISTICAL OR QUANTITATIVE THINKING -- A FUNDAMENTAL INTELLIGENCE

Neil Ullman
County College of Morris

Newsletter for the Section on Statistical Education
Volume 2, Number 1 (Winter 1996)

Neil Ullman's article initiates what we hope to be a fairly regular feature of the Newsletter -- an opinion or provocative piece about the teaching of statistics. The editors heard Professor Ullman's talk at JSM last August and thought it would inaugurate this column well. (Eds.)
I suggest there is a basic Quantitative Intelligence which everyone utilizes all the time. Although this by itself is not necessarily a radical idea, I further propose that there exists an informal spoken language of statistics that acts as a foundation to the traditional written form we teach.

We are constantly measuring, estimating, and experimenting -- all without FORMAL statistics. This informal, essentially subconscious, statistical thinking begins from the moment we wake up in the morning and consider how much longer we can lie in bed. It continues all day in a multitude of ways as we decide whether the water in the shower is "the right temperature," how much coffee to put in the coffee pot, or when to leave for work or school. Just as we have learned to speak words without instruction, we act and think quantitatively without realizing it.

Yet what we teach is generally a formal system relying on an assumption that the student has an internalized familiarity with mathematics, especially of algebra, and hopefully calculus. We begin by presuming measurements have already been taken and concentrate on teaching the grammar and syntax of some subset of mathematical statistics. Rarely is there any connection to something that even we as individuals could say is important to our daily activities or an appeal to our instinctive understanding.

I suggest beginning a process to redefine what we call statistics, and offer some ideas:
  1. Statistical thinking is not just mathematics.

    Statistics should not be a vehicle for teaching mathematics or satisfying a "mathematics" requirement. Statistics needs to be recognized as much as a subject of problem definition and measurement as computing of special quantities. Courses need to be expanded to examine the whys and hows we get to the data in the first place, allowing more time for studying the source of the information and devoting more emphasis to the ways we actually encounter "data." All too often the formal statistical aspect of a problem is the least important.

  2. Incorporate informal quantitative thinking into the statistical psyche.

    Restructure our courses to infuse informal quantitative concepts (accepting them as a legitimate part of a college statistics course) along with the formal statistical practices. Before this can become a universal practice we must develop a common "spoken" language so we all understand the principles and can communicate these ideas amongst ourselves.

  3. Recognize that there is a threshold between informal and formal methods.

    Most of our everyday encounters with "data" do not involve or need formal methods. In spite of all of our admonitions about statistics as a survival skill, most people seem to be surviving without us! Frequently the important consequences are so obvious that we can observe them and react appropriately without special effort. However, when differences are small, problems complex, or our ability to measure not adequate -- then we need to begin to use formal systems.

Let me provide a couple of quick examples of rethinking what we do. I teach at a school where about 99% of my students commute by car. I begin the course with a questionnaire which includes asking how far they travel and how long it takes them. No one has a problem providing values. I challenge them to measure the time and distance it takes for their next trip. The results are brought to the class and then we engage in questions of the cause of different results. This leads to an operational definition so we can compare day to day and student to student trips. Then a data collection can, and does, begin in earnest. Class discussion can bring out points like:

  1. The calculations of mean and standard deviation may be useless if the measurement process is not consistent. If one student chooses to begin timing on leaving the house and another begins when the car is in motion, the values are not comparable. This is just one of the many very common problems with real data collection.


  2. The initial approximation might be more than adequate for most concerns. Furthermore, after gaining some insight from today's trip they have a better idea of what to expect the next time. An informal confidence interval evolves as an intuitive estimation.


  3. A shift in how to act begins if we want to compare routes, times of day, directions, day of the week, and so on. It is in these cases that formal statistical methodologies may be called for, although simple graphical techniques may be all that are necessary.

In a similar fashion I explore how students perceive the comfort level in our classroom. I begin by asking "how do you feel," shift to a 5 or 10 point numeric scale that ranges from too cold to too hot, and finally request an estimate of temperature of the room. Just recording this information, along with collecting their responses into a simple table gives an indication of the relative feelings. A display of their response by seating location can quickly signal differences around the room. Finally I pass around a thermometer which gives a "real" measure of the temperature.

Much insight into the conditions and the state of the room can be learned without any highly formalized statistics. However, if we pose a question such as whether opening the door changes the condition of the room, a formal approach involving taking separate samples and computing estimates of the temperature difference may become necessary.

Let us take heed of the Quantitative Literacy efforts at the elementary level. I recently observed a group of first graders do in-class surveys. They gathered the data and individually prepared bar charts. Most incredible, however, was how they were able to verbalize the results and draw conclusions from their data. They presented their findings in ways I struggle to have my adult students do.

We have been teaching a foreign language. As an adult I have failed miserably trying to learn to read foreign languages, especially when taught by memorizing words and learning grammar. But those same children can readily learn to speak other languages, and accelerate their ability to read them.

Let us move away from the concerns about rigor, and "college level" and recognize that we need to rethink what is basic quantitative reasoning and stress that in standard courses. I recall a trip when the most important, but least accessible, thing I needed to know in a particular foreign language was how to ask the question "where is the bathroom?". What is the most critical thing most people really need in becoming aware of the statistical language?

The above is based on a paper presented in Orlando last summer. Copies of the presentation and a longer paper are available.

Neil R. Ullman
County College of Morris
Randolph, NJ 07869
Phone: (201) 328-5716
FAX: (201) 328-1003
nullman@ccm.edu

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