The Pen-and-Paper Algorithm (PAPA)

Introduction

The Pen-and-Paper Algorithm (PAPA) is one of the most important components of the complex "engine" called DYNAMIC LANGUAGE LEARNING. It comes in two modes:

• Introductory Mode (PAPA IntroMode)
• Standard Mode (PAPASM)

The introductory mode is a greatly simplified version of the standard mode.

We normally teach PAPA with reference to learning *** vocabulary *** (in order to keep the subject matter simple and constant) even though it can be used for many other subject matters as well (grammar, factual information, etc).

Therefore when I keep referring to the learning of "items" and these items are always words, I do this to keep the explanations simple and not because I think language learning consists only of learning vocabulary. I know that there is much more to it. DYLL has many components dealing with other aspects of language learning, such as communication skills.

Before you learn the standard mode of PAPA, you have to spend about four weeks to learn, work with and experience the effects of the introductory mode. The introductory mode is less efficient than the standard mode but it is much easier to learn and to explain.

Sequence of items

The difference between the two modes is the sequence in which the items are tackled. The introductory mode treats all items the same, whereas the standard mode spots, and concentrates on, the more difficult items.

Before learning the standard mode, you must be absolutely familiar with how to learn each item, how to use the folding slip (from which the term "Pen-and-Paper" algorithm was derived), the importance of following all instructions to the letter, the importance of carefully timed revisions, the definition of self-cheating and its negative consequences for your speed of learning. All these are things which we will NOT explain again when describing the standard mode.

A description of the PAPA IntroMode is contained in the article on learning vocabular Link xxx ??? . Read it now before continuing here.

Description of the standard mode compared with the introductory mode

PAPA IntroMode

PAPA IntroMode, in a nutshell, runs as follows:

You try each item in succession from Item 1 to Item 10.

If ten items are correct in succession, that exercise is considered "mastered", and you move on to the next exercise. Otherwise you return to the begining of the same exercise. Keep trying Items 1 to 10 in succession until you have mastered them.

PAPA in standard mode (PAPASM)

PAPA in standard mode (PAPASM), in a nutshell, runs as follows:

Each item has a number (from 1 to 10). Before you tackle an item write down its number on your folding slip. Then guess the correct answer. Then slide the folding slip down to reveal the model answer.

If your answer was correct, move on to the next item.

If your answer was incorrect (e.g. 1 letter wrong), draw a circle around the number (the circle marks a mistake, a wrong item), cross out the wrong answer and copy the model answer, and try to learn something from your mistake. Then move on to the next item.

As soon as three circles are visible on your folding slip, it means that you have identified three difficult items in the exercise. This is not a sign of failure but a sign of success. Identifying a problem is always a success. Only problems which have been identified can be removed. That's why cheating in DYLL is so counter-productive. Give yourself a chance to make your mistakes. They display your weaknesses before they become dangerous.

Your efforts will now be concentrated on these three items. You will continue practising these three items until at least one of them has turned out to be correct.

Intensive care

This is how you work on the circled items. When 3 circles are visible, you copy the circle numbers into the next line on your folding slip. This tells you which items you are now targeting.

Fold the paper back so that your previous answers are no longer visible, but the circle numbers are visible.

Slide the folding slip up (or down) to the first circle number. Try it (ie guess the answer and write down your guess). Then the next item and the next. If three circles are still visible, continue by the same rules, always the same 3 items. Until at least one of these items has been learnt, i.e. fewer than 3 circles will be visible in intensive care.

When fewer than 3 circles are visible, move on to the next item in the main exercise, i.e. try to find another difficult item. A "difficult item" is one where your guess is wrong.

As soon as 3 circles are visible again, apply the 3-circles procedure.

When you have reached the end of the exercise, go back to the beginning. Gradually all difficult items in the exercise will have been identified, and all will have been intensively treated in the 3-circles engine (intensive care unit).

Continue doing this until you can answer 10 items in succession in the main exercise without making a mistake. Then this exercise has been "mastered", and you can move on to the next exercise.

General remarks about PAPASM

This is PAPASM in a nutshell. This description will gradually be refined during the next few weeks. So come back to the website and look for changes.

We will publish more formal specifications of the procedure in the form of a flowchart and several "cycle diagrams". By studying the various forms of explanation, you will eventually be in no doubt about how to use the procedure and its algorithmic character (unambiguous, exact and effective).

We will also explain in detail the psychological reasons why this procedure is so extraordinarily effective, by dynamically adjusting the revision intervals to your performance. (For theory, see the article "Dynamic Learning Algorithms".

For now, in a nutshell, consider the following:

The First-Step method: Divide and rule

If you were told to learn 5,000 words of a foreign language (the sort of vocabulary you need for reasonable competence in a language) you would be downhearted and overwhelmed. How will you ever learn that, especially as you are convinced anyway (wrongly, of course) that you have no talent for language learning.

But if I tell you to learn 500 (to start with), you will think this much more managable and tackle the task with a certain amount of confidence.

Forget the fact that, once you have learnt the first 500, I will give you a second batch of 500 words. Just think of the first 500 and that this is definitely managable.

Scaling that task down has made it easier for you and increased your confidence. This is as if you needed to walk from Leicester to Delhi, because you cannot afford the bus fare. The task might seem impossible. But taking the first step, just one step, is not impossible. So you do it: one step. Then take another step. If you keep doing it, then eventually you arrive either in Delhi or in paradise, whichever comes sooner, and either will be a delight.

So now I make it even easier for you. I divide the 500 words into exercises of 10 words each. That is 50 exercises. Forget about 49 of them. All I am asking you to do is to master one of them, 10 words. Surely you can manage that. Take the whole day if you want to.

That one exercise is "the first step". You have already experienced, with the PAPA IntroMode, that it is easy enough.

You notice that, the smaller I make your task, by breaking it down into its components (Cartesian principle: link), the more feasible it becomes, and your confidence increases ever more.

In the introductory mode, a ten-item exercise is the smallest task you are confronted with. You keep working on 10 words in succession until you have mastered them.

Now imagine that you are still rebelling against that, and tell me you can't do it. Learning 10 words is asking too much, is more than you are capable of achieving.

All right, now PAPASM comes into play. I tell you: You don't have to learn 10 words, just learn three of them, surely you are capable of learning three words.

OK, let's find three words which you don't know yet. To do that, we start the exercise, and try item by item until three circles are visible. Now we have found the three items we want to learn (focus on).

So don't fret. I am not asking a lot of you. Don't worry about having to learn ten words. You don't have to learn ten words. I am a fair man. All I am asking you is to learn three words.

I am not even asking you to "learn" them. I am not asking you to "remember" them.

I am only asking you to ***guess*** them. I can't say no fairer than that, can I? But I can: there is no end to my generosity. I ain't asking you to guess *** three *** words, I am only asking you to guess *** one *** word. Is that a deal? Is that easy enough for you? Will that still be a strain on your patience? Only one word!

If you guess right, wonderful. If you guess wrong, I show you the model answer.

Then we do something to kill some time (i.e. guess two other words) and, about 20 seconds later, I'll ask you to guess the first word again. And so on, no learning required, just a guessing game, with the same word coming back every 20 seconds until you have guessed it correctly.

Once that has happened there will no longer be 3 items in the intensive care unit (but only 2), and you go back into the main exercise to find another candidate for intensive care. Like a talent scout, for crying out loud! Once you have found one, the 3-item guessing game can start again. That's how easy it is! Easy enough even for you?

IntroMode versus Standard Mode

Now let's look at the 10-item exercise in that respect.

You are alternating your activities between the 10-item exercise and the intensive care unit (with a capacity of only 3 items; 3 beds).

What the Intro-Mode and the Standard Mode have in common is that in either mode, all you have to do is to guess and to copy. You are not required to "learn", whatever that may mean. Learning is the automatic result of guessing and copying. There is absolutely no stress in that, and anybody can do it. It is just a game with very precise rules.

The difference between the Intro-Mode and the Standard Mode is that in the Intro-Mode there is no intensive care unit (ICU).

Therefore, in IntroMode, if you make a mistake in, say, Item 3, it will take about 2 minutes before that item is presented again for another guess. The chances of you remembering what you copied last time round are not as great as they would be if that item was presented for guessing 20 seconds after you had copied the model answer.

By opening the intensive care unit we reduce the interval between each of your guessing attempts at the same item. You guess it wrong, you copy the model answer (and try to learn something from your mistake), 20 seconds later you try to guess again, you get it wrong again, etc etc, sooner rather than later you will start guessing correctly. Learning has begun.

Items can hop in and out of intensive care. That is normal. You are not responsible for that, and you don't care. Just follow the rules. Play the game. The results for you are guaranteed. They come when you least expect them -- as a byproduct of your mechanical activities.

Human memory has its own laws, and we exploit them to make our learning as efficient as possible. But there are absolute limits to what is possible, and there is no reason to fret about these limits. For some people, some items require more revisions, or revisions at closer intervals. All we are doing in DYLL is to approximate these intervals, as best we can, to make YOUR learning as efficient as possible. Other people have other requirements and timings. We are concerned only with yours. That's why these algorithms have also been called "adaptive algorithms".

Progress in the exercise as a whole

Looking at the exercise as a whole, the following progression is typical.

Let's assume that you started by copying the words from your textbook into the DYLL Workbook, in the standard DYLL format (each item numbered, numbers in a separate column, question above the answer, blank lines between question and answer, each exercise consisting of 10 items).

While copying these words, some of the easier ones will inadvertently have slipped into your memory and come back to you when you start the guessing game.

If you use the IntroMode, the following might happen:

First round:	8 mistakes
Second round:	6 mistakes
Third round:	4 mistakes
Fourth round:	2 mistakes
Fifth round:	2 mistakes (no progress; some very resistant items remaining)
Sixth round:	0 mistakes. Initial learning finished. Mastery achieved.

The chances are that normally it will work much faster (i.e. fewer rounds required), but this is the sort of progression you have to expect and cope with. One of the greatest obstacles to successful language learning is that adult students (and the older they become the worse it gets) have unrealistic expectations of themselves, get upset, angry with themselves, and AS A RESULT, perform increasingly badly.

One of the major tasks in teaching the techniques of DYLL is to combat the negative attitudes of students. That's why I have to spend so much time arguing against these harmful attitudes which I have come across hundreds of times when teaching this system.

Standard Mode compared to IntroMode

When using the Standard Mode, the difficult items are first identified, and then brought more closely together. Say, in Intensive Care, your repeated guessing attempts are 20 seconds apart. So you make your way very very slowly through the first round of the exercise, with frequent sessions in intensive care.

Obviously, whatever you learn in intensive care may be forgotten again 1 minute later. That is normal and no reason to be upset since the algorithm takes counter-measures. Eventually every item will be remembered for 4 months. DYLL has algorithms which ensure this.

The effect on the exercise as a whole might be as follows:

First round:	8 mistakes
Second round:	4 mistakes
Third round:	1 mistake
Fourth round:	0 mistakes. Mastery

This also might be achieved faster in real life, but I don't want to promise too much. Let your real results outstrip my promises.

What comes after PAPA?

Then the Retention Algorithm (which controls revision times and dates) comes into play to stretch the retention span from 15 minutes to 4 months.

Any items which slip through the net of the Retention Algorithm will be mercilessly passed to the Enforcer Algorithm (extraordinary rendition), which makes sure that even the most difficult, most recalcitrant, most obstinate, most perverse, most rebellious item is brought to heel, is tamed like Shakespeare's Shrew and peacefully joins your repertory of obedient servants.

Then the various communication algorithms come into play and teach you how to utilise your linguistic tool box in real life.

This is a first attempt at describing the Standard Mode of PAPA. The psychology behind it is explained in "Dynamic Teaching Algorithms". Various different ways of presenting the same procedure will be posted here soon.

Real-life examples:
A student learns with PAPASM

I have preserved some of the folding slips I used when learning Urdu numerals so that you can see what these slips look like. You can see how many mistakes I made, how often I repeated the same mistake, how an item which I got right initially turned out wrong at a later attempt. All this is normal - but traditional learning methods are so slack and woolly that there is not even a record of what is happening, and therefore nothing useful can be learnt from it. You can also see how my performance improved during my early revisions.

In this case it was the numbers which had to be translated into Urdu. Therefore, on these slips, there is no distinction between the item number and the question. In this case, the item number *** is *** the question.

To make sure that you can see where the paper was folded, I have drawn a line along the fold. In real life, this is not necessary. You just fold when the system requires it.

Enforcer Exercise x
followed by Exercise 11, initial learning

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Now you get to see the real thing, warts and all.

The following slip deals with two exercises, first an Enforcer Exercise, a selection of numbers which I found particularly difficult, particularly resistent to learning. To enable you to trace my progress, I am listing here the numbers appearing on the slip.

Enforcer Exercise: 80, 50, 76, 78, 89, 57, 97, 87, 88 mistake, 36 -- fold and start again -- 88 mistake again within seconds of previous correction, 50, 76, 78, 89, 75, 57, 97, 87, 88 (88 correct this time, about one or two minutes have passed since previous correction) -- mastery achieved for complete exercise, 10 items correct in succesion (for once). This difficult exercise obviously had been revised repeatedly on previous days. Otherwise the first round would have contained many more mistakes. The Enforcer Algorithm will ensure that this exercise is repeated again and again with only a few days' distance between each revision. Mastery today does not mean mastery after several days.

Exercise 11, Initial Learning (R0)

After having completed this exercise, I went on immediately to the next due Exercise, Exercise 11, using the same folding slip, and even continuing in the same line. A fold was not necessary because the next exercise contained different numbers (or words).

This was the first attempt at Exercise 11 (Initial Learning). Not too many attempts during the first round since these numbers had already been discussed and practised in class and I had seen them while doing my analysis of the number system. If you were confronted with these numbers for the very first time, I would predict 9 or 10 mistakes during the first round, because these numbers are hellishly difficult to remember.

The teacher, of course, thought we knew these numbers since he had discussed and practised them with us in class. These slips prove that the teacher, like many teachers, was far too optimistic concerning our skills. Teaching us something does not mean that we know it.

You notice from the slips that follow that the exercise contains several difficult numbers which sometimes come out right, then wrong, then right again, i.e. they have not been reliably learnt. In DYLL we want reliable learning, and whatever is not known reliably is treated as not known, and eventually passed to the Enforcer in whose realm there will be howling and gnashing of teeth.

Exercise 11, first official attempt, proceded as follows: 49, 50, 51, 52, 53, 54 mistake, 55, 56, 57 mistake, 58 mistake -- 3 circles visible, fold paper. Note down three items to be done: 54, 57, 58. Intensive Care starts: 54 wrong again about one minute after last correction, 57, 58 -- only one item remains with circle, i.e. only one item is still in intensive care. We therefore return to main exercise to find more items which merit intensive care. We copy number 54 with circle to indicate that it is already a candidate for intensive care once we have found two more patients. 49, 50, 51, 52, 53 mistake, 54, 55 mistake -- three circles visible, 2 patients moved to intensive care, write their numbers into the next line. Fold the slip. -- Note three intensive care numbers: 54, 53, 55. Start work: 54 wrong again, this is obviously a stinker. See how often this number is circled on the slip. But that is no reason for getting upset. I trust in the Enforcer and simply continue calmly following the rules.

The slip is now full. New slip started, and the same exercise is continued on it.

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Three items go into intensive care and will be practised in the order in which they arose: 54, 53, 55. Work starts: 54 wrong again, 53, 55 -- return to main exercise and continue with next number in sequence (after 55 comes 56). Work in main exercise starts: 56, 57, 58. End of exercise. Therefore circled number, 54, must be copied with its circle into the
next line to indicate that it is still on the waiting list for intensive care. Paper is folded. Work starts at the beginning of main exercise: 49, 50, 51, 52, 53, 54: 54 is now correct without recourse to intensive care. 10 correct responses in succession have been given (from 53 to 54). Therefore mastery has been achieved. This exercise is passed to Retention Algorithm. I can stop working or proceed to the next exercise.

Exercise 11, R1: after 15 minutes

The first revision of Exercise 11 is due after 15 minutes. This is the most important of all revisions (see discussion of Retention Algorithm and the article "Dynamic Learning Algorithms" for the reasons).

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These are the results of Revision R1, after 15 minutes. 10 items correct in succession, including the stinker, 54. So I could translate the numbers one by one after 15 minutes, which does not mean that I could have done so after 60 minutes or after one day. It also does not necessarily mean that I can recite these numbers. Some of the correct answers on the slip came after some hesitation and were successful guesses. This is not matured knowledge by any means.

But we must be aware that *** all *** answers, even those based on assumed knowledge, are guesses. The difference between guesses is that some are made with more confidence and others with less confidence, but even those made in the pub with the greatest confidence turn out to be absolutely wrong.

Then there are guesses which have a greater probability of turning out right than others.

In this exercise I described those answers as "guesses" where of two options the chances of one or the other being correct were about equal in my mind; I just didn't know, but I did know it had to be one or the other; e.g. pan/van, or pan/ppan.

However, the fact that I did get them right is, I presume, evidence that I am moving *** towards *** true knowledge. Right guesses are better than wrong guesses. People with more knowledge are more likely to guess right than people with less knowledge.

Mine will be true knowledge only when I can guess with 99% certainty that a certain numeral has to be, say, "cau-van" and not "cau-pan". DYLL will ensure that I reach this state. That is the target, and this is the state of mind in which I want to step into any examination hall -- when I would be quite happy to argue with any examiner that he is wrong and I am right. That is firm knowledge.

Exercise 11, R2: after 60 minutes

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After 1 hour, the next revision of Exercise 11, R2, was due. Here are the results.

After one hour, I scored 80% correct (2 mistakes). This is below the expected score of 90%. During the second round I made a mistake with 50 (which I got right during the first round). It was a stupid mistake which I should not have made, an accident (Flüchtigkeitsfehler in German). I should not have doubled that "c". During the second round I simply did not pay attention. I was so cocksure about that word.

This is no excuse as far as the DYLL rules are concerned. A mistake is a mistake, and the prescribed measures have to be taken. If I had known that word as thoroughly as I know that 2 * 2 = 4, I would not have made that mistake even if I had been completely inebriated. Therefore this mistake does indicate a weakness which has to be eliminated.

However, the fact that I got this word right during the first round shows that I had some real knowledge, and that the memory model still held quite good during this revision.

Exercise 11, R4:
first thing next day (= after 12 hours)

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We omit the record of Exercise 11, Revision R3, last thing at night on the same day, and show now the folding slip for Revision R4, first thing the following morning. The results were better than predicted: 100% correct after 12 hours. (100% must never be predicted but it can often happen.)

This success does, of course, not mean that the battle has been won. I can not yet say that I "know" these numbers, as students naively tend to say. I can only say that I "know them with a retention span of 12 hours", or I can say that I "have passed a certain test". Tests and knowledge are related, but they are different things.

That retention span has to be stretched, gradually and systematically to 4 months with the help of the Retention Algorithm in tandem with the Enforcer Algorithm and a number of random techniques (e.g. Hindi-Urdu Roulette).

Exercise 9, R0: Initial Learning

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Notation for these comments:
(59) means: a mistake was made in this item and it was circled on the folding slip.
60+ means: this item was answered correctly.
In the reminder: (59) means: 59 appears in a circle, i.e. it is on the waiting list for Intensive Care.

- -----------------------------------------

Main Exercise starts: (59), 60+, (61), (62). Three circles visible. Write the three circled numbers into a fresh line. Fold.

Reminder says: 59, 61, 62. No circles are needed in this reminder to indicate waiting list status since Intensive Care will start immediately for these three items.

Start intensive care: 59+, 61+, 62+

Continue in Main Exercise: 63+, (64), (65), (66). Three circles visible. Write the three circled numbers into a fresh line. Fold the paper.

Reminder says: 64, 65, 66.

Start Intensive Care: 64+, (65), 66+.

Continue in Main Exercise: (67), 68+.

End of exercise reached. Only two circles visible. Copy the two circled numbers, together with their circles, into a fresh line, to indicate that these two numbers are on the Waiting List for Intensive Care until a third candidate is found. Fold the paper and start again at top of exercise.

The performance so far (many mistakes) is typical for an exercise which contains many unknown items and where the student is doing a lot of guessing into the dark. This was the first pass. Watch the improvement taking place in the second pass, as a result of the intensive care work that has been done. Observe, however, that the second pass is not, and should not be expected to be, error-free.

Reminder says: (65) (67).

Main Exercise, second pass starts: 59+, 60+, 61+, 62+, 63+, 64+, 65+

We now cross out the encircled 65 in the last Reminder, since this mistake has now been "fixed". Therefore only one circle is visible at this stage, namely (67).

Main Exercise, work continues: (66), (67). The slip is now full, and work continues on the second slip. Starting a new slip is the same as folding a slip. Not only are we at the end of an exercise, but also three circled numbers are now visible: 67, 66, 67, and these items now go into Intensive Care.

These items count as three items even though 67 is a duplicate. The DYLL rules apply rigorously, and they make sense! 67 needs even more Intensive Care treatment than a normal Intensive Care patient because 67 has been identified as particularly difficult: The student got it wrong a second time, after an interval of only a few seconds. That means that the revision intervals for this item have to be *** radically *** reduced. Having this item twice in the same Intensive Care list means that it will be tried and tried again with in tiny intervals: less than 20 seconds retention will be tested. That's what's required for extremely difficult items.

In extreme cases, a single item can appear three times in the same Intensive Care list. The effect gives that the student has to copy this item three times in succession, something that DYLL very rarely prescribes. If that does not solve the problem, we call on the "Enforcement Procedures" and atomise the item before rebuilding it again.

Exercise 9, R0: Initial Learning,
continued on next folding slip.

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Reminder says: 67, 66, 67. Intensive Care starts: 67+, 66+, 67 plus. Fold slip.

Main Exercise continues: 68+. End of exercise reached. Fold slip and start again at the beginning.

59+, 60+, 61+, 62+, 63+, 64+, 65+, 66+, 67+, 68+. End of exercise reached. 10 items correct in succession. Mastery achieved.

Exercise 9, R1: after 15 minutes

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All items correct during first pass, as predicted. No detailed comment required.

Exercise 9, R2: after 60 minutes

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Main Exercise starts: 59+, 60+, 61+, 62+, 63+, 64+, 65+, 66+, 67-, 68+

90% correct after one hour. That is good, as expected. Write the circled number together with the circle into a fresh line. Fold the slip. Move to top of exercise.

Reminder says: (67)

Main exercise starts: 59+, 60+, 61+, 62+, 63+, 64+, 65+, 66+, 67+

Mistake 67 has been eliminated and the circled number is crossed out. The student has given 10 correct responses in a row (from 68 to 67). Therefore he has satisfied the conditions for mastery and can leave this exercise, even though he has not yet reached the end of the exercise.

DYLL demands 10 items correct in a row because these items provide a certain measurement of retention time (say, one or two minutes). By taking this hurdle, the student proves that he has remembered the last "difficult item" for at least as long as it takes to make 10 responses in succession. That's the purpose of this DYLL rule.

Exercise 9, R4: first thing next day

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No comment required.

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