Why was x2.0 from corner to corner chosen for our Standard Chart?

[Answer contributed by Dr. Ogden Lindsley, 17 September 2000.]

This question is more complicated than a simple why. Our standard chart was not flippantly designed overnight. It resulted from fifteen years of full time thinking and much careful testing in practice.

In 1955 at my Harvard Behavior Research Laboratory I found we needed at least a 4 cycle multiply chart for daily tracking human behavior. When my chronic psychotic patients accelerated out of the 0 to 70 responses per hour add scale on their daily charts we had to rechart on a 0 to 700 scale. When they accelerated out the top of that we recharted on a 0 to 7000 per hour scale. A few accelerated out of that and we recharted on a 0 to 70,000 per hour scale. This told me we needed at least a 4 cycle multiply scale up the short side of a standard chart. All commercial charts had their multiply scales up the long side. I was so busy grant hustling and drug screening that I never took time to design and custom print a standard multiply chart that I knew we needed.

From 1965 through 1967 I knew we needed 6 times ten multiply cycles for the full spread of human behavior frequencies, from 1 a day to over 300 a minute. My students' dissertations of that time show that I had not yet decided on a standard time scale across the bottom which determines the corner to corner celeration or learning angle value.

Tom Caldwell's 1967 telecoaching dissertation contained eight 6 cycle by 25 day charts. Eric Haughton's 1967 tailoring consequences dissertation contained one 3 cycle by 9 minute, four 6 cycle by 9 minute, one 3 cycle by 40 days, and three 6 cycle by 40 days charts. Ann Duncan's 1967 comparison of rates of gifted with normal children dissertation displayed three 6 cycle by category, and three 6 cycle by 6 cycle correlation charts. The data for these dissertations were collected in 1965 and 1966. We had not yet chosen standard chart time dimensions and a standard celeration angle.

Carl Koenig, Joe Edwards, and I measured weekly celerations in around fifty acceleration and fifty deceleration projects done by our educational research group. We found the maximum acceleration was x8 and the middle x1.4 per week. Our largest deceleration was /7 with a middle of /1.4 per week. We recharted twenty five studies published in behavior therapy and special education journals and found their middle acceleration was also x1.4 and their middle deceleration was also /1.4. This meant we must design our standard charts to display x1.4 and /1.4 celeration angles clearly.

I decided to set corner to corner at x2 and /2 to provide a celeration or learning aim above what we were then accomplishing at x1.4 and /1.4 per week. Keeping the frame at overhead projector and computer screen size meant that our time scale would be 140 days or 20 weeks. This gave us about one school semester on each chart sheet. Joe Edwards' 1969 dissertation contained 40 standard 6 times ten multiply cycle by 140 day charts. Edward's charts were collected from a learning disability classroom in the fall of 1967.

Doubling and halving have been deep in the heart of our culture for hundreds of years. This strengthened my choice of x2 per week. All living cells double their number by dividing in half. Children have shouted, "Double or nothing," and "I double dare you" in their games over the years. Our language has the words half-baked, half-hearted, and half-assed. The ancient game of backgammon has a doubling cube to indicate the current value of the stake as a result of doubling. An ancient Persian legend tells of the vizier who invented chess choosing as his reward a grain of rice on the first square, two grains on the second square, and doubled from there on to the sixty fourth square which bankrupted the kingdom. The sixty fourth square would have 16 followed by 18 zeros, or 16 quintillion grains of rice! An equally ancient French legendary question asks about a lily pad in a pond that doubles every day. When the pond is half covered, how many days will it take to fully cover the pond with lily pads? One day!

Doubling time has been used for over a century in economics and future studies to describe and compare growth. World population doubles every 40 years (2.1 % increase per year, x1.021 celeration per year, x1.110 celeration every 5 years). Bank deposits at 7% and world fertilizer consumption double every 10 years (7% increase per year, x1.07 celeration per year, x1.403 celeration every 5 years).

Half-life has been used by physicists to describe the rate of radioactive decay for about a century. Physicians also use half-life to describe the rate of decay of drugs in our bodies. After two half-lives one quarter of the radioactivity is left, proving that half-life is really just divide by two. Radium's half-life is 22 years (3.2% decrease per years, /1.032 celeration per year, /1.171 celeration every 5 years). Uranium's half-life is 269 x (10)5 years.

The 12,000 different projects on 1,223 different behaviors stored in our Behavior Bank computer supported our choice of x2.0 (Lindsley, Koenig, Nichol, Kantor, and Young, 1971). Carl Koenig's 1972 dissertation demonstrated that our standard chart would accurately straight line project 5 to 7 frequencies 50% of the time.

Chung-Jung Liao's 1984 thesis collected 176 standard charts published in two books, two journal issues, and four volumes of the Journal of Precision Teaching. Chung-Jung found a celeration spread of x10 to /5 per week. The middle acceleration was x1.4 and the deceleration /1.4. Kathy Porter's 1985 dissertation recharted 576 percentage interval records from the Journal of Applied Behavior Analysis from 1968 through 1984. Kathy found middle before celerations for both acceleration and deceleration targets of x1.0 with a spread of x18 to /100 per week. The middle during celerations were x1.0 for acceleration targets and /1.2 per week for deceleration targets. Deborah Ehling's 1986 dissertation recharted 640 frequency or number charts from the Journal of Applied Behavior Analysis from 1968 through 1984. Deborah found middle during celerations were x1.0 for 352 acceleration targets and /1.3 per week for the 288 celeration targets.

The celerations on our standard charts from the Behavior Bank and these dissertations supported our choice of corner to corner to be x2.0 per week, an angle of 34 degrees.

References

Caldwell, T. E. (1967). Can pupil performance rates tell us when a student teacher is ready for her own class? Unpublished doctoral dissertation, University of Kansas, Lawrence.

Duncan, A. D. W. (1967). Behavior rates of gifted and regular elementary school children. Unpublished doctoral dissertation, University of Kansas, Lawrence.

Edwards, J. S. (1969). Precisely teaching children labeled learning disabled. Unpublished doctoral dissertation, University of Kansas, Lawrence, KS. Dissertation Abstracts International, 1970, 34, 5162A. (University Microfilms No. 70-11. 017).

Ehling, D. G. (1986). Standard Celeration Chart summary of Applied Behavior Analysis effects. Unpublished doctoral dissertation, University of Kansas, Lawrence, KS.

Haughton, E. C. (1967). A practical way of individually tailoring classroom consequences. Unpublished doctoral dissertation, University of Kansas, Lawrence.

Koenig, C. H. (1971). The behavior bank: a system for sharing precise information. Teaching Exceptional Children, 3(3), 157.

Koenig, C. H. (1972). Charting the future course of behavior. Unpublished Doctoral Dissertation, University of Kansas, Lawrence.

Liao, C. (1984). A quantitative review of published Standard Celeration Charts (1970-1983). Unpublished master's thesis. University of Kansas, Lawrence, KS.

Lindsley, O. R., Koenig, C. H., Nichol, J. B., Kanter, D. B., & Young, N. A. (1971). Handbook of precise behavior facts: Listings of the first twelve thousand published precise behavior management projects. Kansas City, Kansas: Precision Media. 2 volumes.

Porter, K. L. (1985). Standard Celeration Chart summary of individual-percentage-interval recording studies from the Journal of Applied Behavior Analysis 1968-1984. Unpublished doctoral dissertation, University of Kansas, Lawrence, KS.