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Monday, June 29, 2009

Why Standards Fail: An Open Letter to OAG and Other Standards Committees

I wrote this letter in response to an article I had read in in some computer magazine decrying the failure of some standard which had been championed by the author. I actually don't recall which magazine or which standard. I don't think it was ever published. As you can see, I have been quite interested in successful standards design for quite some time.



Why Standards Fail
An Open Letter to OAG and Other Standards Committees.
By Grant Robertson
(written sometime between 1993 and 1996)

Why is it that, often, even good standards fail while, at the same time, obviously inferior ones thrive. Perhaps it is because those creating the standards forget three very important factors for success in the world today: Marketing, marketing, and, of course, marketing. What many don't realize is that a standard is a product that must be sold like anything else. It must be sold to the vendors who will implement it and it must be sold to the customers who will pay for it. What good does it do to make the worlds biggest mouse trap if there are no mice that big? And how do you expect to sell the world’s best snargthik traps if no one would know what a snargthik was even if they saw one.

It is the same with standards. You could create a huge, voluminous standard. But if it has to be implemented on an all or nothing basis then most vendors won’t touch it, or will only implement random parts of it, and most customers wouldn’t want to pay for it if they did. Why spend a fortune on software that adheres to some new standard if it’s going to fall flat because no other vendors adopted it? And why pay any money at all for software that does something no one has heard of or thinks they’ll ever need just because it does so in a standardized way.

Sunday, June 28, 2009

Importing Microsoft Word Documents Into Blogger

I have a lot of old papers that I decided to post here on my blog. They are all Word 2003 documents and many of them have quite a bit of formatting which I did not want to replicate by hand in HTML. So, I have tried a bunch of weird tricks and figured out a system for importing Word documents, preserving the formatting in the post while avoiding messing up the CSS styles in the rest of the blog page. This is a relatively convoluted procedure. However, once you try it a couple of times it will work relatively quickly, at least compared to reformatting a 20 page research paper by hand.

One of the steps in this procedure is made a little easier if you have Adobe DreamWeaver. I use CS3 but I don’t know for sure which older versions have the features you will need. It is not absolutely necessary and I will explain a workaround at the appropriate place. Also, I use Microsoft Windows XP. Part of this procedure depends on the behavior of the Windows clipboard. I cannot guarantee that it will work in Mac OS or Linux. However, some of the CSS editing tricks will work for anyone.

Basic Procedure

  1. Copy and paste the Document from word into the Blogger editor compose tab.
  2. Isolate the imported Word document by surrounding its HTML code with a <span class="UniqueName"> and </span> set of tags in the Blogger editor’s HTML tab.
  3. Fix the Word CSS stylesheet by inserting "span.UniqueName " before all of the CSS selectors (without the quotes but with the trailing space), thereby creating descendant selectors that will only apply within the imported Word document and will not affect the formatting of the rest of your blog.

Detailed Procedure

  1. Open the Word document.
  2. Open the Blogger editor to the HTML tab.
  3. If you want to type some introductory text, explaining a bit about what the paper is about or why you wrote it then do it in the Blogger editor now. Surround each paragraph with <p></p> tags.
    • If I do this step then I also put a horizontal rule <hr/> under my initial comments and before the actual paper or Word document I intend to import.
  4. Prepare the Word Document:
    • Often academic papers will have been formatted with double line spacing. Usually this does not read well online so you will want to remove it. If the spacing is set in the styles then you should edit the styles rather than simply selecting all of the text and setting it to single line spacing. The latter results in Word placing a style code in each and every major tag in the document.
    • Text boxes get exported as pictures so, if there are any of these in your document that need to pull the text out of the text box and place it where it will be appropriate within the main body of the document.
    • Remember, all you have to do is not save the modified Word document and these changes will not hurt anything.
  5. Select all of the text in the Word document that you want in your blog post and copy it.
  6. Go to the Blogger post editor and switch to the Compose tab.
  7. Place the cursor under any introductory text you may have entered in Step 3 and paste the Word document there.
    • When you do this paste either Word or the Windows Clipboard will automatically convert the Word formatting to CSS styles and assign HTML element class names to each of the paragraphs which will be enclosed in <p> tags.
    • If you now switch to the Blogger editor’s HTML tab you will be able to see quite a lot of extraneous HTML elements. Some we will keep and some we will either get rid of or ignore. Part of this mess is the CSS stylesheet for this particular document. You will notice that it is now contained in what will be the body of your final blog web page rather than within the <head> tag as is normally the case. This is apparently OK. It is also what will allow your post to keep the same formatting as the original document.
    • If you were to publish your post at this point it is entirely possible that some of the CSS styles listed in Word’s CSS stylesheet would conflict with the CSS stylesheet in your Blogger template. The next steps will rectify that situation.
  8. Clean up extraneous HTML tags.
    • If you have DreamWeaver you can use it to modify the HTML that resulted from the previous paste.
      1. Copy all the HTML from the Blogger editor HTML tab and paste it into a new DreamWeaver HTML document between the <body> and </body> tags in the code editor (NOT the design view).
      2. In DreamWeaver choose { Commands / Clean Up Word HTML… ; <basic> } and select the following check boxes:
        • Remove all word specific markup
        • Clean up <font> tags
        • Fix invalidly nested tags
        • Apply source formatting
        • (Do NOT select “Clean up CSS.”)
      1. Click [OK].
      2. Select all of the HTML code between the <body> and </body> tags in the DreamWeaver code editor.
      3. Copy that and paste it in place of ALL the HTML code that is in the Blogger editor’s HTML tab.
    • If you do not have DreamWeaver you can safely ignore the extraneous HTML tags if you want. Or you can edit out whatever bits that Blogger complains about when you try to post later. I have found that anything between and tags can go. Also all of the meta tags can go. This is what Blogger will complain about the most.
  1. Fix the CSS so that Word’s CSS styles do not conflict with Blogger’s CSS styles.
    • You may not even have to do anything for this step. Most of the HTML element class names and associated CSS selectors created by Word have unique names that will be very unlikely to conflict with the CSS in your Blogger template. In fact, if all of the styles in your Word document were ones you created yourself then it is almost certain that none of their names will conflict. However, if you simply used the Normal style or any of the heading styles without creating your own, uniquely named, version of them then you will probably have problems. This is because Word sometimes uses a simple p, h1, or h2 element selector in the CSS styles. Since most Blogger templates use the <div> tags with specific element id attributes to associate HTML elements with CSS styles, the simple p or h1 selectors in the Word CSS stylesheet takes precedence.
    • There is an easy way to check for these problems. Simply publish your post then look at your blog. Check to see if any of the formatting of the text outside of your post has changed. Check both on your blog’s home page and on the post’s individual page. If nothing is wrong then you are done.
    • If some of the CSS styles have conflicted then follow these steps:
      1. Go to the Blogger editor’s HTML tab and surround all of the HTML code in the post with a <span class="UniqueName"> and </span> set of tags.
        • This sets the post off with it’s own unique class that will apply to the entire imported Word document but not to any of the rest of the blog page or any of your other posts. If you entered introductory text in Step 3 then you may want to place the first <span> tag just after that text and the optional <hr/> tag. This will cause that introductory text to retain the same styling as the rest of your blog, thereby visually setting it apart from the text imported from Word.
        • You must use a unique class name for each of your posts so that one post won't interfere with another.
      2. Look for Word’s CSS stylesheet near the top of the post.
        • If you were able to use DreamWeaver to clean up the HTML then it will be right at the top of your HTML code (or right under your introductory text from Step 3). If you are choosing to just ignore the extraneous HTML then the part you want may be buried a little bit from the top. It will be between a <style> and </style> tag and NOT within any and tags.
        • You will notice that this CSS stylesheet is not formatted nicely at all. All the CSS rules are just strung together on one line of text.
      3. Insert "span.UniqueName " before each of the CSS selectors.
        • Do not insert the quotes but make sure to include the space between the inserted text and the existing selector.
        • This creates what they call a descendent selector. This means that the simple selector will only apply if it is also found buried somewhere within the specified ancestor element. In this case this means that the simple element selector will only apply to elements that are inside our <span class="UniqueName">and</span>set of tags, which means only within our imported Word document.
        • Remember to insert the "span.UniqueName " before each selector in grouped selectors separately or it will only apply to the first one. (Grouped selectors are a list of selectors separated by commas.)
        • Word puts HTML comments around its CSS stylesheet. In DreamWeaver this makes it all light gray. It is safe to remove the HTML comments from around the CSS rules within the <style> tags. This allows DreamWeaver to use its syntax highlighting which makes it much easier to find all of the CSS selectors.

Pictures

If your word document has any pictures in it then you will have to do “a few” additional steps. Again, it goes pretty quick once you get used to it.

  1. In Word { File / Save as Web Page… ; File name: = “whatever.htm” ; Save as type: = “Web Page (*.htm, *.html)”[v] ; [OK] }.
    • This is just to get the pictures out of the file quickly and easily. You should not use the resulting HTML code to paste into Blogger. For some reason copying and pasting from Word into Blogger’s Compose tab cleans up a few things that are difficult to clean up any other way. (At least with my limited knowledge.)
    • Word will create a folder and stick all the files in it.
    • You can just save this to the desktop because you will be deleting it when you are through.
  2. Go through the image files that Word created and rename them to something more meaningful.
  3. Upload all these images to your preferred file or image hosting service. You could just use Picassa if you want.
  4. Go through all the HTML that was the result of pasting the document into Blogger and edit the <img> tags one by one.
    1. Locate the uploaded image on your image hosting service.
    2. Right click on it and choose “Copy Image Location” or some similar menu item.
      • You want to copy the image’s URL to the clipboard. Sometimes copying the URL from the browser’s address bar does not give the correct URL. Always get it by right clicking on the image itself.
    3. In the appropriate <img> tag select all of the text between the quotes for the src attribute and paste the new URL there.

Notes

  • If you look at the source for this page you will see that it was originally created in Word. While I could have created it directly in DreamWeaver, I wanted you to be able to look at the code as an example.
  • I have found that it is easier to do all of this editing in DreamWeaver. I just copy all the HTML from Blogger’s HTML tab and paste it between the <body> tags in a new DreamWeaver HTML file. Then it is easy to resize the images if necessary without guessing at widths and heights. If I have to resize the image appreciably (which may be necessary if your blog’s main <div> is pretty narrow as in many templates) then I will then turn that image into a link to the full size version of the image for the user’s convenience. All you have to do is paste that same URL into the link field in the image properties bar in DreamWeaver and it automatically wraps the image in an <a> tag. When I am finished I just copy all the HTML code between the <body> tags and paste it into Blogger’s HTML tab, replacing the previous code.
  • In some of these instructions I have used Grant's Concise GUI Notation System (GCGUINS) which I have described in a separate blog post.
  • I should warn you that things do not always come out peachy keen.
    • If you have a deeply indented hierarchical list in your Word document then some levels of it will get turned into HTML lists and some of them will simply be turned into paragraphs with <p> tags instead.
    • If you have any sample HTML tags in your document (like I have lots of in this one) then pasting the text to Blogger's Compose tab will not convert those to appropriate HTML entities (&lt; or &gt; etc.). You will have to dig through the HTML code and switch them around. The easiest way to do that is to cut them from the code window in DreamWeaver and then paste them in the Design window. DreamWeaver will then convert them to the appropriate entities in the HTML code so they show up properly in the post.
    • If you subsequently edit your post in the Blogger editor's Compose tab then it may strip out some or all of the formatting. So, once you have done this you should always use Blogger's HTML tab or an external HTML editor to edit the post rather than the Compose tab.
  • I am sure that others will find many other problems with this technique. Sometimes it goes quickly and sometimes, as with this particular document, it takes a lot of additional editing. Only you can decide whether it will be faster to use this technique or to completely reformat your document in HTML. Feel free to post comments here or in the Blogger Help Forum about any other problems you find.
  • Even if you don't copy and paste from Word some of the other tricks here could help.
    • Using the <span class="UniqueName"> and <span> tags around the entire post to isolate it and then using span.UniqueName to create descendent selectors in your CSS will allow you to use different formatting in your post.
    • If you no longer have access to the original pictures used in your document then the technique shown here can get them out quickly and easily.

This post is Copyright © 2009 by Grant Sheridan Robertson.

Friday, June 26, 2009

A Brief History of the Mathematical Definition of Forgetting Curves

This is the final version of the term paper I wrote for my Psych 301 class, "The Psychology of Learning." It is shorter and more narrowly focused. It is 13 pages, not including the title page and references. It does have an extra section about the emergent nature of the power function. So, I shaved off about 4 pages and added one. I have not combined these two papers because I want these posts to accurately reflect what I actually turned in for a grade.


A Brief History of the Mathematical Definition of Forgetting Curves
by Grant S. Robertson
Written July, 2008

When attempting to investigate learning, 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 retained appears to decrease over time. (Ebbinghaus, 1885) 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 retention under various conditions. In 1885, Ebbinghaus published a book wherein he described how he performed his tests and even explained his method of statistical analysis.

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. 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. 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, 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). 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.

The Discovery of Forgetting Curves and Early Research Supporting the Concept

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 a book of his research called, “Memory: A Contribution to Experimental Psychology” and what a contribution it was. In nine chapters, Ebbinghaus discussed the qualitative nature of the, then current, thinking on the subject, then moved on to provide details on the experiments he performed. 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. Then, depending on varying conditions, attempt to relearn that same list to the point where he could repeat it twice again. 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, 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.” “Obliviscence” simply means “Forgetting” (A Dictionary of Psychology, 2001 cited in encyclopedia.com, 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). 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 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 forth 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 began a study wherein she used Chinese symbols to test retention. Rather than learning to recite the symbols or recall them when questioned, the Wylie study merely tested whether participants 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). 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. 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.

A. R. Gilliland tested to see if the Ebbinghaus forgetting curve applied to more real world material. Gilliland used a series of picture cards depicting a complex office scene. Participants were given 30 seconds to study the image and then were immediately asked to recall as much as possible. The quantity of material recalled was used as a baseline for later trials. Gilliland found that initial retention did not fall off as quickly as Ebbinghaus had observed. 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). 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 explored in the context of witness viability (Deffenbacher, Bornstein, McGorty, & Penrod, 2008). In their 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).

Finding a Mathematical Definition Through 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. 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 and 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 vs. Calculated

Notice how closely the calculated values follow the observed values with the exception of the forth data point discussed earlier.

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 (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 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(1960). 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 an accurate 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, p. 133). 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 to which brain cells do not adhere. 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 curve 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, admitted that the forth 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.

“The Power Law” as an Emergent Property

An “emergent” property is a property or characteristic that appears only when other more fundamental properties or aspects of a phenomenon are combined. For instance, each water molecule in a lake has a mass and velocity, along with the quantum forces that keep the molecules from getting too close or far away from each other. It is the combined movement of all those individual molecules in a lake or ocean that create the waves and currents that we see. In the same way, it has been shown that the Wickelgren Power Law may likely be the result of a combination of many more fundamental phenomenon acting in concert.

Drawing on the work of Rubin and Wenzel (1996), Sverker Sikström of the University of Toronto used advanced mathematical modeling and simulation to show that power functions appear when multiple different exponential curves are combined. He made what he claims are biologically sound assumptions: that learning rates will vary slightly over time, that individual parts of the system (e.g. brain cells) will have varying effects, or “weights” within the system, and that those weights will have “bounds” or limits on their total effect within the system. Sikström concluded that the power function is a natural result of the combined effect of all the brain cells participating in a network involved in learning a particular thing. These individual brain cells behave in an exponential manner but their combined activity produces the emergent property of the power function for the network as a whole (1999). Additionally, Sikström makes the flat statement in his abstract that “Empirical forgetting curve data have been shown to follow a power function” (1999, p. 460).

Soon after Sikström presented his paper, Richard Anderson of Bowling Green State University did additional work showing that the power function could appear emergently from the combination of many different more fundamental functions. Anderson also simulated the forgetting process using a computer. However, he used range-limited linear functions, range-limited logarithmic functions, basic power functions as well as basic exponential functions as the assumed function for each individual simulated brain cell in the network. He found that the power function emerged most strongly when the variability between the different “cells” in the network was highest in almost all cases. The only exception was when the basic function was already a power function, in which case the simulation always resulted in a power function regardless of variability. Anderson concludes by speculating that the power law, rather than being ubiquitous because it is a fundamental biological property of brain cells, is ubiquitous simply because it the mathematical result of combining many different response curves with a high degree of variability between them (2001).

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,” a preponderance of the articles that were discovered seemed to take “the power law” as a given and use it as the basis for further research. In addition, the power law may not be a fundamental property of the nature of learning. However, as a rather ubiquitous emergent property, it will likely be possible to use it as the basis around which to devise learning and education systems.

References

Anderson, R. B. (2001). The power law as an emergent property. Memory & Cognition, 29(7), 1061-1068. Retrieved from www.csa.com

Brichacek, V. (1971). Mathematical models of learning: II. Ceskoslovenská Psychologie, 15(2), 144-157. Retrieved from www.csa.com

Boreas, T. (1930). Experimental studies on memory. II. the rate of forgetting. Praktika De l'Académie d'Athènes, 5, 382 ff. Retrieved from www.csa.com

Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354-380. Retrieved from http://content.apa.org/journals/bul/132/3

Chappell, M. N. (1931). Chance and the curve of forgetting. Psychological Review, 38(1), 60-64

Deffenbacher, K. A., Bornstein, B. H., McGorty, E. K., & Penrod, S. D. (2008). Forgetting the once-seen face: Estimating the strength of an eyewitness's memory representation. Journal of Experimental Psychology: Applied, 14(2), 139-150

Dictionary of Psychology, A. (2001). Retrieved July 2, 2008 from http://www.encyclopedia.com/doc/1O87-obliviscence.html

Ebbinghaus, H. (Ruger, H. A. & Bussenius, C. E., Trans. 1913). Memory: A contribution to experimental psychology. New York, NY, US: Teachers College Press. (Original work published 1885)

Gold, J. M., Murray, R. F., Sekuler, A. B., Bennett, P. J., & Sekuler, R. (2005). Visual memory decay is deterministic. Psychological Science, 16(10), 769-774

Lane, S. D., Cherek, D. R., Lieving, L. M., & Tcheremissine, O. V. (2005). Marijuana effects on human forgetting functions. Journal of the Experimental Analysis of Behavior, 83(1), 67-83

MacLeod, C. M. (1988). Forgotten but not gone: Savings for pictures and words in long-term memory. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14(2), 195-212

Murdock, B. B. Jr., & Cook, C. D. (1960). On fitting the exponential. Psychological Reports, 6, 63-69.

Roediger, Henry L.,III (1). (1985). Remembering Ebbinghaus. US: American Psychological Association

Rubin, D. C., & Wenzel, A. E. (1996). One hundred years of forgetting: A quantitative description of retention. Psychological Review, 103(4), 734-760

Sikström, S. (1999). Power function forgetting curves as an emergent property of biologically plausible neural network models. International Journal of Psychology.Special Issue: Short-term/working Memory, 34(5-6), 460-464

Wickelgren, W. A. (1974). Single-trace fragility theory of memory dynamics. Memory & Cognition, 2(4), 775-780. Retrieved from www.csa.com

Wixted, J. T., & Carpenter, S. K. (2007). The wickelgren power law and the ebbinghaus savings function. Psychological Science, 18(2), 133-134

Wylie, M. (1926). Recognition of chinese symbols. American Journal of Psychology, 37, 224-232


This post is Copyright © 2009 by Grant Sheridan Robertson.

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.

History

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 encyclopedia.com, 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.

References

Austin, S. D. M. (1921). A study in logical memory. American Journal of Psychology, 32, 370-403. doi:10.2307/1414001

Brichacek, V. (1971). Mathematical models of learning: II. Ceskoslovenská Psychologie, 15(2), 144-157. Retrieved from www.csa.com

Boreas, T. (1930). Experimental studies on memory. II. the rate of forgetting. Praktika De l'Académie d'Athènes, 5, 382 ff. Retrieved from www.csa.com

Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354-380. Retrieved from http://content.apa.org/journals/bul/132/3

Chappell, M. N. (1931). Chance and the curve of forgetting. Psychological Review, 38(1), 60-64. doi:10.1037/h0075176

Deffenbacher, K. A., Bornstein, B. H., McGorty, E. K., & Penrod, S. D. (2008). Forgetting the once-seen face: Estimating the strength of an eyewitness's memory representation. Journal of Experimental Psychology: Applied, 14(2), 139-150. doi:10.1037/1076-898X.14.2.139

Dictionary of Psychology, A. (2001). Retrieved July 2, 2008 from http://www.encyclopedia.com/doc/1O87-obliviscence.html

Dietze, A. G., & Jones, G. E. (1931). Factual memory of secondary school pupils for a short article which they read a single time. Journal of Educational Psychology, 22(8), 586-598 & 22(9), 667-676 (in 2 parts).

Ebbinghaus, H. (., Ruger, H. A. (. (., & Bussenius, C. E. (. (1913). Memory: A contribution to experimental psychology. New York, NY, US: Teachers College Press. doi:10.1037/10011-000

Gold, J. M., Murray, R. F., Sekuler, A. B., Bennett, P. J., & Sekuler, R. (2005). Visual memory decay is deterministic. Psychological Science, 16(10), 769-774. doi:10.1111/j.1467-9280.2005.01612.x

Lane, S. D., Cherek, D. R., Lieving, L. M., & Tcheremissine, O. V. (2005). Marijuana effects on human forgetting functions. Journal of the Experimental Analysis of Behavior, 83(1), 67-83. doi:10.1901/jeab.2005.22-04

MacLeod, C. M. (1988). Forgotten but not gone: Savings for pictures and words in long-term memory. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14(2), 195-212. doi:10.1037/0278-7393.14.2.195

Murdock, B. B. Jr., & Cook, C. D. (1960). On fitting the exponential. Psychological Reports, 6, 63-69.

Roediger, Henry L.,III (1). (1985). Remembering Ebbinghaus. US: American Psychological Association. doi:10.1037/023895

Rubin, D. C., & Wenzel, A. E. (1996). One hundred years of forgetting: A quantitative description of retention. Psychological Review, 103(4), 734-760. doi:10.1037/0033-295X.103.4.734

Wherry, R. J. (1932). The curve of forgetting: Its statistical application. Journal of Educational Psychology, 23(8), 621-624. doi:10.1037/h0070645

Wickelgren, W. A. (1974). Single-trace fragility theory of memory dynamics. Memory & Cognition, 2(4), 775-780. Retrieved from www.csa.com

Wixted, J. T., & Carpenter, S. K. (2007). The wickelgren power law and the ebbinghaus savings function. Psychological Science, 18(2), 133-134. doi:10.1111/j.1467-9280.2007.01862.x

Wylie, M. (1926). Recognition of chinese symbols. American Journal of Psychology, 37, 224-232. doi:10.2307/1413689


This post is Copyright © 2009 by Grant Sheridan Robertson.

Why hast thou forsaken me?

Why hast thou forsaken me?: An Analysis of Green Day’s “Jesus of Suburbia”

By Grant S. Robertson

Written June 2005


At over nine minutes long and in five clearly delineated sections, Green Day’s “Jesus of Suburbia” is more of an anthem than simply a song. The second track on the latest album, Idiot America, – from one of America’s most popular and longest surviving bands with gutter punk roots – one might assume this composition is nothing more than another rant against standard American culture and society. Commonly believed to hate all of American culture and everything it stands for, this segment of society – large enough to fill venues across the country – is too important to ignore. In The Closing of the American Mind Allan Bloom laments that America ’s youth use modern music to isolate themselves from the rest of the world (68) and that this isolation will leave them deaf to the rest of the “great tradition” of culture that is available to them (81). But is it a two way street? Could it be that society is giving young people plenty of reason to plug in and tune out? If adults invested a little time to tune in and discover what this music is saying, perhaps a bridge could be built between the generations.

A quick glance at an album cover or even reading through some lyrics may not reveal all that is available to us. As Henry Louis Gates Jr. wrote in his 1990 article "2 Live Crew, Decoded: Rap Music Group's Use of Street Language in Context of Afro-American Cultural Heritage Analyzed", there are often coded languages that develop within cultures or sub-cultures. A form of coded language, not discussed by Gates though, appears as a sub-culture within almost every culture in the world. This is the language of music; what the voices, tones, and rhythms are telling the listener at an almost subconscious level. Sweet melodic tones mean happiness, low tones played slowly convey a hint of sadness, while even lower discordant tones often symbolize anger. Reading the codes within both forms of language can be an extremely powerful tool to help us bridge this gap of understanding.

So who is this new Jesus? Well, he’s certainly not perfect (IV.9) having come from a broken home (V.14) that was potentially violent as his claim to be “…the son of Rage…” (I.1) seems to indicate. His reference to “Brad’s (sic)” (I.14) could even be slang for the never ending series of boyfriends that some single mothers subject their children to. He feels that he doesn’t really fit into society; belonging, instead, to the “None of the Above” (I.3) category. On top of that he is kept hyped up on soda pop and calmed down by Ritalin (I.4) hinting that his caregivers don’t really accept him for who he is. When he tells us that, “No one ever died for my sins…” (I.5) he is letting us know that no one has really done anything of significance to help him either. So he sits around zoning out on television (I.12) and drugs. (I.16-17) But there is some hope of salvation for this new Jesus. He knows “there’s nothing wrong with…” (I.8) him and he is exactly “…how [he is] supposed to be,” (I.9) at the very least.

At just over thirty, with children of his own, Billy Joe Armstrong, the writer and lead vocalist for Green Day, can’t be our guy. He doesn’t fit the bill. Maybe he did during his own youth but not any more. So, again, who is this new Jesus? With more than thirty references to “I”, “me”, or “my” spread throughout the lyrics one might feel safe in assuming that this Jesus was a single individual; and a pretty self centered one at that. Subtle clues, however, point to the possibility that this “I” may be a collective. Throughout the song, various musical styles are used to convey special meaning. In this first section a standard pop rock beat, the most popular by far according to Allan Wells’ report "Popular Music: Emotional Use and Management" (108) gives us our clue that this Jesus may be a rather large collective. More than a “we” or a group of individuals, this “I” speaks with one voice and one mind.

A collective Jesus has to live somewhere even as he/it exists everywhere. The title is quick to point out that our Jesus lives in Suburbia. This is backed up by references to 7-11 parking lots (II.2) and shopping malls (II.14) where suburban kids tend to hang out. What better place for a collective Jesus to live than a collective location in a “… town that don’t exist” (V.5) “at the center of the earth” (II.1). Everywhere and nowhere.

The world has not been kind to our collective Jesus either. Though it hasn’t crucified him or thrown his disciples to the lions it has done something far worse but much more subtle. After all the suffering the first Jesus endured leading up to and during his crucifixion, he only complained about one thing. Many Christians believe that when Jesus cried out, “My God, my God, why have you forsaken me?” (Matthew 27:46) it signifies that his God has left him alone for the first time in his life. To be forsaken or denied acceptance can be the cruelest thing one being can do to another. And our new Jesus certainly feels forsaken by the entire world. He was taught a motto that turned out to be a lie (II.3) and when he followed the signs misleading him to nowhere he found himself in the City of the Damned (II.10). A lost child with no one who really cared (II.11-12). So, in the final analysis, it seems our new, collective Jesus just happens to be the disenfranchised youth of suburban America .

A common theme among these young people is that they just don’t seem to care about anything. The way they dress in sloppy clothes, the vacant stare on their faces, the ever present headphones playing music that often carries the same message: “I don’t care!” Green Day’s “Jesus of Suburbia” even has an entire section specifically devoted to it. However, a deeper analysis reveals something entirely different. The first line is actually the phrase, “I don’t care if you don’t.” repeated many times. Speaking to those who have abandoned him, our collective Jesus sings this line in a sing-song, rather mocking voice accompanied by music that is energetic and danceable. If played at a night club or rave this would be the section that gets people jumping up in the air. When people dance around while speaking in these mocking tones you generally get the idea that they don’t mean what they are saying.

The last phrase of this line reveals how the Jesus truly feels. After the mocking tones his voice suddenly shifts into a low growl. “I don’t care!” (III.1) he screams, accompanied by low guitar riffs and plenty of distortion. This shows how incredibly angry he is; and you don’t get that angry when you really don’t care. The entire rest of this section is sung in this same angry tone with a voice distorted to sound almost demonic. Even the handwritten lyrics for this section in the liner notes are written using a bold marker with letters almost four times larger than all of the rest of the song. It seems he is intentionally protesting far too much in order to drive home the point that he really does care. Being the central section of the song, perhaps it is also the central message: The disenfranchised youth represented by this new Jesus do care and it is really hurting them to be considered worthless outsiders by the American culture just for being different.

So what does our disenfranchised collective Jesus do? What’s the solution to his problem? Not being mature enough to deal with his problems rationally, he simply runs away (V.3&6). Not to a completely different city; but to the parking lot of the nearest 7-11, to the shopping mall, and to his headphones. Anywhere away from the family and the world that has rejected him. In this isolation within a crowd he listens to music which, on the surface, repeats exactly how he feels while subliminally saying what he can’t quite find the words to say himself: “Please just accept me for who I am.”

While the disenfranchised youth of suburban America may not be taken seriously by their more hardcore peers, the gutter punks; are being ignored by their parents; and pretending that they just don’t care about any of it; their choice in music sends a different message. They aren’t buying the cultural motto being pushed out to them. They do feel disenfranchised, ignored, or even forsaken just for being different and believing an alternate set of truths. They don’t hate the American culture but they do hate the way they are treated by it. Finally, despite their protestations, they do care and it hurts them to be considered worthless outsiders just because they are different. So their solution is to shut down and tune out. American culture needs to pay attention as this can become a problem in and of itself. As more and more youth in all segments of society are turning to escapism through drugs or music they are less involved in providing the solutions that they themselves will need in only a few short years.



Works Cited


Bloom, Allan. "Music." The Closing of the American Mind. New York: Simon, 1988, 68-81.

Gates, Henry Louis Jr. "2 Live Crew, Decoded: Rap Music Group's Use of Street Language in Context of Afro-American Cultural Heritage Analyzed." New York Times. 19 June 1990: A23.

Green Day. “Jesus of Suburbia.” American Idiot. Reprise Records, 2004

Solomon, Jack. "Masters of Desire: The Culture of American Advertising." Signs of Life in the USA : Readings on Popular Culture for Writers. 4th ed. Ed. Sonia Maasik and Jack Solomon. Boston: Bedford/St. Martin's, 2003. 120-171.

Wells, Alan. "Popular Music: Emotional Use and Management." Journal of Popular Culture. 24.1 (1990): 105-117.


This post is Copyright © 2009 by Grant Sheridan Robertson.