Thursday, June 25, 2009

A Brief Synopsis of the Long History of Forgetting Curves: A Review of the Progression from Initial Discovery to Mathematical Definition

This post is a draft of the paper I wrote for my Psychology 301 course, "The Psychology of Learning" which was really about behaviorist theory. I did the research and wrote the paper in about three weeks during the last half of the summer term. I wasn't taking any other courses but I was installing windows in the house at the same time, if that counts for anything. The teacher said it was too long for the final paper and asked me to narrow the focus and trim it down considerably. It is 16 pages, not including the title page and references. I have included the entire draft just in case it might be helpful to someone.

A Brief Synopsis of the Long History of Forgetting Curves: A Review of the Progression from Initial Discovery to Mathematical Definition
by Grant S. Robertson
Written July 2008

When attempting to investigate learning – whether it be the acquisition of Pavlovian associations, behaviors operantly conditioned, or facts studied purposefully – it is unfortunate that scientists are as of yet unable to take direct measurements, within the brain, of how much has actually been learned at a given point. Indirect measures must instead be used to determine what has been learned, or – more importantly – what has been retained. As is common knowledge, the amount of material retained appears to decrease over a period of time. (Ebbinghaus, 1885) This, it can be imagined, has been known since humans first became conscious of their ability to remember things at all. For most of history, this fact has been either not given much thought or discussed in purely qualitative terms. However, in about 1879 Herman Ebbinghaus began a six-year research project where he devised tests to quantitatively measure the amount of material retained under various conditions. In 1885, Ebbinghaus published a book with nine chapters and a preface – though, surprisingly, no conclusion – wherein he describes in great detail how he performed his tests and even explained his method of statistical analysis.

In his work, Ebbinghaus discovered that retention falls off quickly at first and then more slowly later in a pattern that has since been referred to alternatively as a "forgetting curve," "forgetting function," retention curve," or "retention function." In the 123 years since the publication of Ebbinghaus's work there has been much research done on the nature of these "forgetting curves" and what affects them. Some of the work appears to be merely academic while some was designed for the constructive purpose of determining how to improve methodologies for teaching humans or training animals. After firmly establishing that the amount of material retained in memory does, in fact, follow a curve which is steeper at first and shallower later (e.g. Austin, 1921; Wylie, 1926; Boreas, 1930; MacLeod, 1988), researchers turned their attentions to determining an exact equation to express the nature of this curve (e.g. Wherry, 1932; Murdock, 1960; Wickelgren, 1974; Rubin & Wenzel, 1996) in a mathematical process called “curve fitting.” Different primary mathematical functions have been attempted, from Ebbinghaus’s first logarithmic function (1885) to exponentials (Murdock, 1960) and power functions (Wickelgren, 1974). After over a hundred years and thousands of research projects, one enterprising pair of researchers undertook the arduous task of performing a curve fitting analysis for over 210 different data sets matching each of them to 105 different two-parameter functions (Rubin & Wenzel, 1996). After much analysis, they determined that three different basic functions performed equally well. A further review of what has become known as the “Wickelgren Power Law” (Wixted & Carpenter, 2007) reveals that it not only can be made to fit the existing data but can accurately predict later data-points when given only the first few. As any scientist will tell you, the ability to predict future results is the calling card of a good theory.

Finally, the establishment of any scientific theory is not of much value unless it can be put to use. Therefore, this review ends with a very brief exposition of what can be accomplished using accurate and predictive forgetting curve functions along with some suggestions for future research.


Herman Ebbinghaus (1879 – 1885)

In 1879 Herman Ebbinghaus began his landmark study attempting to quantify memory which he defined “in its broadest sense, including Learning, Retention, Association and Reproduction” (Ebbinghaus, 1885, Preface). In 1885 he published an entire 128 page, nine chapter book detailing his research called, “Memory: A Contribution to Experimental Psychology” and what a contribution it was. In the nine chapters, Ebbinghaus discussed the qualitative nature of the, then current, thinking on the subject, then moved on to list the myriad exhausting experiments he performed and how he gathered his data. In his experiments, Ebbinghaus – using himself as his only subject – would memorize a list of nonsense syllables to the point where he could repeat them twice without assistance. Then, depending on varying conditions, attempt to relearn that same list to the point where he could repeat it twice again without assistance. This technique was devised to remove ambiguity from his research. Rather than simply asking (himself) how many items from a list he could remember – which he determined to be an unreliable means of measuring retention – he used the savings in time necessary to relearn the lists as a measure of how much had been retained. He used these techniques to test for retention as a function of a variety of different conditions: the length of the list, the number of repetitions, simple time since the original learning, repeatedly learning the same lists at different times, and as a function of the order of the items in the list, each described in its own chapter.

The chapter most important for this review is “Chapter 7. Retention and Obliviscence as a Function of the Time.” For those not familiar with 19th century German translated into early 20th century English, “Obliviscence” simply means “Forgetting” (A Dictionary of Psychology, 2001 cited in, 2008). In this experiment, Ebbinghaus attempted to determine exactly how much he could remember after a given period of time (§ 27). In order to avoid the confounding variable of practicing the same list over and over again, he learned 163 different lists with 13 syllables each and only attempted to learn each list twice: once as the original learning and once at a predefined time after that original learning. He divided the 163 lists into eight different groups and attempted the second learning after a different delay for each group (§ 27). During the course of this work, Ebbinghaus discovered that it took him much longer to learn or relearn lists during the evening so he devised an adjustment factor for those trials that took place in the evening (§ 27). But then, who can blame him after learning 163 different lists of 13 nonsense syllables? So, after lots of exhausting tests and a bit of adjustment for different conditions, Ebbinghaus arrived at the following simple chart depicting how much he tended to retain after a specific period of time:

Ebbinghaus Chart 1

Though this chart was obtained from the web-based publication of a translation of the original work, it is believed to be a direct scan from a copy of that translation. Column I depicts the delay between the original learning and the relearning trial for each of groups 1-7. (The eighth group was the control group which he simply used to determine how long it took to learn a similar list in the first place.) Column II is the amount of material retained as a percentage. Column III is the “Probable Error” as explained in Section 10 of Ebbinghaus’s book. Column IV is simply 100 minus the Q% from column II.

Though Ebbinghaus discussed the “curve” produced when these figures are plotted, he never produced an actual plot of the data. Following is a plot of this data created in Microsoft Excel. The smooth curve in the plot is a result of the smoothing feature of the software. Only the actual data points from the table above are of importance.

Ebbinghaus Curve 1

You may notice that the fourth data point is slightly higher than one might expect if memory behaved in a smoothly progressing fashion. Ebbinghaus discusses this in Section 29.2 and attributes it to possible errors in his methodology or the exercise thereof. All-in-all, this plot of the first “forgetting curve” in history is still quite telling. It reveals how retention falls off quickly at first and then shifts to falling off quite slowly. From the curve seen above it is difficult to ascertain whether the actual amount of material retained would drop all the way to zero or would asymptote on some positive, non-zero number.

Groundbreaking as it was, Ebbinghaus’s study was not without its critics. One-hundred years after the publication of Ebbinghaus’s book, Henry L. Roediger summarized these criticisms thus, “First, he employed only one subject-himself. Second, and a more common criticism today, is that the artificiality of Ebbinghaus's experimental conditions guaranteed that nothing important or useful could be found from his research. His research and the tradition it spawned is alleged to lack external validity.” (Roediger, 1985) Though, Roediger’s criticisms do have merit, this author believes it is a bit of an overstep to claim that the entire tradition spawned by Ebbinghaus lacks external validity. The work of Ebbinghaus was a starting point. In the over 100 years since, a great deal of good research has been done to expand on what Ebbinghaus started and to bring it into the real world through practical application.

Follow-up Studies on Forgetting Curves

In 1917 Margaret Wylie, of Cornell University began a study wherein she used Chinese language symbols to test retention. These symbols, although meaningful, were of no real meaning to the subjects (save one Chinese girl). Rather than learning to recite the symbols or recall them when questioned, the Wylie study merely tested whether subjects could recognize the symbols they had seen in the past. Wylie found that the ability to recognize symbols followed the same type of curve that Ebbinghaus observed for relearning nonsense syllables (Wylie, 1926).

In her 1921 study, Sarah D. Mackay Austin did a follow-up study to determine if the Ebbinghaus forgetting curve was apparent for logical or real-world material as it was for nonsense syllables. She had subjects learn real world facts that they did not already know but would be of interest to them, then she tested them for retention. She did not use Ebbinghaus’s more accurate method of measuring the savings in relearning the material and instead simply asked the subjects to recall or answer questions about what they had learned. Still, she obtained forgetting curves that were similar to those of Ebbinghaus (Austin, 1921)

A Greek researcher, Th. Boreas published a study in 1930 in which he replicated Ebbinghaus’s work with the addition of using verses instead of nonsense syllables and over a much longer period of time. He found that retention for verses parallels that of nonsense syllables although the slopes in both the short and long terms are shallower than that observed by Ebbinghaus. In one case Boreas observed that absolutely none of the original learning for nonsense syllables could be detected after a 10 month period (Boreas, 1930) This may indicate that the forgetting curve does asymptote to zero after all.

The 1948 research of A. R. Gilliland was also designed to determine if the Ebbinghaus forgetting curve applied to more real world material. In this experiment Gilliland used a series of picture cards depicting a complex office scene. Subjects were given 30 seconds to study the image and then were immediately asked to recall as much as possible. This was used as a baseline against which to compare later attempts to recall the material. Gilliland found that initial retention did not fall off anywhere nearly as quickly as Ebbinghaus had observed in his personal experiments. Gilliland concluded that Ebbinghaus was far too pessimistic in his estimation of how fast retention initially falls off and attributed this to Ebbinghaus’s use of nonsense syllables rather than real world material (Gilliland, 1948) However, it must be noted that the Gilliland study is also flawed in several ways. First, the simple act of attempting to recall material from the cards in order to obtain the baseline, thereby reinforced those particular facts in the minds of the subjects and therefore confounded the findings of later recall. Second, a complicated system of points based on the subjects’ confidence levels in their answers was used. This could easily have been mis-engineered such that it positively affected the retention scores that were subsequently graphed. Third, recall of objects in a picture is quite a bit less difficult than recall of nonsense syllables or even recall of other, more real world, facts. Finally, though recall of what one observes in the real world does often come up in real life, it is not an essential skill for learning the types of material that one must learn in school or to understand the world around oneself. We are not given pictures of historical events and then later asked if there was a dog in the corner of the picture for our final grade. We must learn who was in the picture and why. The actual appearance of the picture is not a priority. Therefore, it can be claimed that Gilliland’s study was as far from reality as that of Ebbinghaus. In the end, though, the results Gilliland obtained did reflect more rapid fall off of retention nearer to the original learning than later, thus confirming the crux of the work Gilliland contends to refute.

The forgetting curve has also been studied in many different real world contexts. This includes, but is not limited to: retention of facts from reading material by secondary school subjects (Dietze & Jones, 1931). Wherein it was found that this meaningful material in a meaningful context still follows the Ebbinghaus forgetting curve, albeit in a form that is quite stretched out over time. The researchers attribute this to the meaningfulness of the material rather than to some deficit in the Ebbinghaus work. The phenomenon has also been explored in the context of witness viability (Deffenbacher, Bornstein, McGorty, & Penrod, 2008). In this meta-analysis of 53 other studies, the researchers determined that retention and recognition of human faces fell off at a rate that matched that described by the Wickelgren Power Law, a function which accurately describes the Ebbinghaus forgetting curve and will be discussed later in this review. Other contexts that have been studied are remembering pictures (MacLeod, 1988), visual memory decay (Gold, Murray, Sekuler, Bennett, & Sekuler, 2005), and even marijuana use (Lane, Cherek, Lieving, & Tcheremissine, 2005).

Curve Fitting

Once it had become firmly established that the curve of forgetting did, in fact, follow the same path first described by Herman Ebbinghaus in 1885, researchers began to look for an equation – with a theory to back it up – which accurately expressed the nature of this forgetting curve while allowing for all the variations due to context and material that had been observed over the years. As psychology researchers attained greater mathematical skills and, later, as computer software became available to make the job easier, researchers devised various equations and used curve-fitting (a mathematical process by which parameters of an equation are adjusted until the equation most closely matches the available data) to test these equations against their data.

Ebbinghaus made the first attempt at curve fitting, though he did not call it that, in his book. He devised a logarithmic equation that closely fit the data from his original experiment.

b = 100k/((log t)c +k)

where b is percent retained,

t = time since original learning, and

c and k are constants:

k = 1.84

c = 1.25 (Ebbinghaus, 1885, § 29.3)

He then created the chart comparing the observed and calculated values shown below:

Ebbinghaus Chart 2

In order to make these numbers easier to understand a plot of them has been created in Microsoft Excel and is presented below:

Ebbinghaus's First Forgetting curve with Calculated Values Overlain. (They are almost identical.)

Notice how closely the calculated values follow the observed values with the exception of the fourth data point discussed earlier. The mathematics behind these calculations are more thoroughly explained in “The curve of forgetting: Its statistical application” by R. J. Wherry of Cumberland University, Lebanon, Tennessee (1932).

Further attempts at finding a mathematical expression to define the forgetting curve were hit and miss for a while, as were the theories that accompanied them. Matthew N. Chappell of Columbia University presented an interesting theory that because learning involves a transfer of energy and that energy begins being dissipated right from the beginning then the forgetting curve must necessarily follow a logarithmic function (Chappell, 1931). After extensive searching of the CSA Illumina aggregated database of research articles using search terms such as “AB=forgetting and AB=(curve* or function or functions) AND AB=(mathematical or (curve NEAR (fit or fitting))” no further attempts to express the forgetting curve mathematically could be found up until 1960. At that time, Murdock and Cook (1960) published “On fitting the exponential” where they endeavored to educate the Psychology community about the mathematical methods of curve fitting using the exponential as an example. Then, in 1971 a Czech researcher did a review of attempts to mathematically model learning from 1962 to 1971 (Brichacek, 1971) however the full text of the article was not available. This reviewer can only surmise that either not much thought was given to finding this function up until the 1960s or that earlier researchers merely assumed the Ebbinghaus formula was correct.

The Wickelgren Power Law

Wayne Wickelgren, according to Wixted and Carpenter, “studied the time course of forgetting more assiduously and more effectively than anyone since Hermann Ebbinghaus” (Wixted & Carpenter, 2007). In 1972 Wickelgren published a major study which included his Strength-Resistance Theory, a mathematical theory based on the logarithm which he claimed accounted for many different aspects of the forgetting curve and how it varies depending on subject matter and context. Wickelgren also presented a considerable body of research illustrating how well his theory matched the data (Wickelgren, 1972). However, just two years later he published his groundbreaking “Single-trace fragility theory of memory dynamics” (Wickelgren, 1974). In this theory, Wickelgren refuted the notion that short-term memory is separate from long-term memory and instead proposed a mathematical theory which encompassed them both. This theory claims that a memory “trace” (a term used by Ebbinghaus (1885, § 26)) can be described by a series of equations that include both an exponential and a power function. A mathematical explanation of these equations is outside the scope of this review (see Wickelgren, 1974, 775-776). For those of us who are not experts in calculus and differential equations, a better – though still not entirely simple – explanation of the math behind this theory can be found in Wixted and Carpenter’s “The Wickelgren Power Law and the Ebbinghaus Savings Function” (2007).

One of the main tenets of this new theory is that the forgetting curve is affected by two primary parameters: the strength of the memory trace and its fragility. Strength refers to the degree of learning associated with a particular memory. Whereas fragility refers to the difficulty the mind will have in retaining that memory over time. If one studied a long series of nonsense syllables until one could recite them easily multiple times the memory trace would likely be characterized by a high degree of strength during the recitations. However it would suffer from a high degree of fragility because there is nothing on which to attach the memory within long term memory and the nonsense syllables have no value for the individual. On the other hand, an important conversation with a loved one might have a memory trace characterized by low strength and low fragility. One might not remember the exact words after just an hour but one would remember the gist of the conversation for perhaps a lifetime. The third parameter is a constant which has yet to be determined. Its purpose is to account for the fact that we measure time in arbitrary units which brain cells do not adhere to. The final parameter is simply the time since the original or most recent learning period.

This reviewer finds Wickelgren’s theory quite compelling not only because it has a simple logical explanation but because it matches the data so very well. In “One hundred years of forgetting: A quantitative description of retention,” Rubin and Wenzel analyzed 210 different data sets and attempted to fit each of 105 different two-parameter equations (1996), not including the additional parameter of time. They made sure to include equations that had been suggested by researchers but also included some relatively random equations just to see what would come up. In the end they presented three different equations as having the most promise: a simple power law, the hyperbola-in-the-square-root-of-t forgetting function (which had never been proposed before), and what they called the “Rubin-Wenzel-Wickelgren-Weibull-Williams-Watts exponential-power law” (p. 758). It is the review of Wickelgren’s work done by Wixted and Carpenter (2007) that is the most convincing. Not only do they clearly explain the logic behind Wickelgren’s theory, but they illustrate how the equation accurately predicts latter data points when only given the first few. As the authors state, this is a much stronger indication of the accuracy of a formula than simply being able to be fit to a complete set of points (p. 133). Below is reproduced Figure 1 from Wixted and Carpenter’s 2007 paper. In this graph, Wixted and Carpenter use the data from the original Ebbinghaus results and attempt a curve-fitting with both the Wickelgren Power Law and a simple exponential function. You may recognize these data points from the Excel plot presented earlier.

Wickelgren vs. Exponential

As the caption says, even when only the first five data points are used in the curve-fitting process the Wickelgren Power Law accurately predicts the last data point even though Ebbinghaus, himself, admits that the fourth data point must be in error (Ebbinghaus, 1885, § 29.2). In fact, when multiple different curve fittings are overlain on one another, it is almost impossible to discern a difference between them. Pay close attention to the thickness of the plot at the top compared to the bottom. It is slightly thicker at the bottom indicating that the plots don’t line up exactly, but are very close. Compare this to the curve fitting of a simple exponential to the same data using the same procedure: first with only the first five data points then with one more for each iteration of the process. Notice how the curves produced (though they look like straight lines) do not even intersect the last two data points as oversized as they are. Also notice how the line moves dramatically with each additional data point that is considered. It should be noted that the simple exponential function was not one of the top three functions selected by Rubin and Wenzel (1996) and that none of the other top candidates were analyzed in this manner by Wixted and Carpenter. It would be interesting to see just how well the other two functions stack up against the Wickelgren Power Law under the same graphical analysis.

Conclusion / Discussion

In the hundred-odd years since Herman Ebbinghaus did his first experiments, using savings in learning time as a measure of retention, and made his first attempts at formulating a mathematical function that could describe the forgetting curve he obtained, much research has been done to both verify Ebbinghaus’s findings and to perfect an equation to describe that curve. Many functions have been theorized, and one has been found to both fit the available data and predict future results accurately. Though no wide-spread, unanimous support was found for the “Wickelgren Power Law,” this reviewer thinks that the theory is strong enough to be the basis for future research.

Looking forward, additional research could be conducted on what is often called “spaced repetition” or the “spacing effect” based on the Wickelgren Power Law. A cursory review of past research on the spacing effect (see Cepeda, 2006) seems to reveal that most researchers are looking for a specific amount of time to delay between repetitions of study in order to maximize learning. This author believes that generally optimum time delays will not be found because the optimum time will be different for each different combination of subject, material, and conditions just as the forgetting curves vary based on those same parameters. Rather than look for generally applicable time delays, the author proposes that the time delays should be based on the specific retention values predicted by the Wickelgren Power Law for that specific combination of subject, material, and conditions. Using vector calculus different points could be chosen along the predicted forgetting curve based on the degree of curl in the curve. In other words, the change in radius of the curve as it transitions from steep to shallow slope. Then experiments could be designed that test which of those points correspond to the optimum time delay for restudy of material with the goal of decreasing the slope of the forgetting curve a certain period of time after the last study period. It is this author’s contention that by basing these repetition delay periods on the forgetting curve itself rather than on arbitrary time delays that far more efficacious methods can be developed for improving learning.


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This post is Copyright © 2009 by Grant Sheridan Robertson.

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