Knowledge structuring and representation in learning based on active recall

Piotr Wozniak, 1994

This text was taken from P.A.Wozniak, Economics of learning, Doctoral Dissertation (1995), and adapted for publishing as an independent article on the Web (P.A.Wozniak, Apr 23, 1997). For an equivalent text written in popular-scientific language see: 20 rules of formulating knowledge for learning

In this article I will attempt to show that it is feasible to dismember learned knowledge into small pieces that can be attributed to individual semantic units of topics in a knowledge system subject to a learning procedure. I also predict the inevitability of such an approach in future self-instruction systems based on active recall.

We will discuss elements that, independently of the repetition spacing algorithm (e.g. as used in SuperMemo), influence the effectiveness of learning. In particular we will see, using examples from a simple knowledge system used in learning microeconomics, how knowledge representation affects the easiness with which knowledge can be retained in the student’s memory. The microeconomics knowledge system has almost entirely been based on the material included in Economics of the firm. Theory and practice by Arthur A. Thompson, Jr, 1989. Some general concepts of macroeconomics have been added from Macroeconomics by M. McKenzie, 1986, while the mathematical capsule items have been taken from Marketing Research by David A. Aaker and George S. Day, 1990.

  1. Knowledge independent elements of the optimization of self-instruction

  2. Knowledge representation issue in learning

  3. Components of effective knowledge representation in active recall systems

  4. Sequencing items in the stepwise process of acquiring associative knowledge

  5. Techniques for minimizing the complexity of synaptic patterns as a key to keeping A-factors high

  6. Planned redundancy as a way to cross-strengthening synaptic patterns

  7. Complexity of wording vs. comprehension

  8. Additional functionality encapsulated in items

  9. Summary of knowledge representation issues in learning


Knowledge independent elements of the optimization of self-instruction

Before I move toward representation of knowledge, I would only shortly like to list some other principles of effective learning that are representation independent:

Knowledge representation issue in learning

It has been known for ages that the way knowledge is represented affects the way it is remembered, and consequently, the degree of how easily it can be retained in memory over a longer period.

The art of mnemonics is as old as the art of learning, and professional mnemonists, with their trained memory capabilities, can truly leave an average mortal speechless in face of top mnemonic feats. Indeed, the mnemonic techniques are very easy to apply, and most of capable students, more or less consciously, use them in their daily routine. However, by conscious understanding of the rules and principles, even the best students can gain a great deal.

The basic principle of mnemonics, from the neurobiological point of view, is to build memory images from as many previously stored engrams as it is only possible. As visual processing in the human brain seems to involve much more sophisticated circuitry than, for example, verbal processing, extensive use of visual imagery is a key to success. Instead of memorizing a nonsensic telephone number, the student can memorize a collection of visual scenes unequivocally mapped to numbers, and generate a unique, easily memorizable sequence of graphic events that can serve as an effective way of representing the number. Recalling the phone number can then be equivalent to invoking the stored visual event and translating it to the sequence of digits, or more often, to a sequence of two-figure numbers. As I will try to argue in later paragraphs, minimizing the number of synaptic connections involved in storing memories is the key to maximizing retention over a longer period. Representing new memories as easily recoverable composites of old memories serves exactly that end.

To simplify the discussion of knowledge representation issues with respect to the complexity of neural connections involved in storing particular engram, I will shortly introduce a concept of a synaptic pattern.

As early as since the introduction of sensitive neural activity measurement techniques, it has been known that memories can be associated with spatiotemporal patterns of synaptic activity, or synaptic patterns in short. There exists a substantial terminology confusion as far as naming the concept of the synaptic pattern is concerned. Therefore, it is worth noticing than in relevant literature, the notion of synaptic patterns, often devoid of its temporal component, might be used more or less synonymously with terms such as: cell assembly, neural structure, synaptic structure, synaptic net, synaptic activity pattern, etc.

As I will try to show in the chapter devoted to biological aspects of memory, the complexity of synaptic patterns is likely to be in strict correlation with item difficulty (e.g. as expressed by the A-factor). It is therefore central to understanding the principles of effective representation of knowledge in self-instruction systems based on active recall and repetition spacing.

Items that do not comply with minimum complexity of synaptic patterns will, in the course of repetition, gradually experience loss of its components. In other words, memory will accomplish natural selection of the core synaptic pattern with elimination of all additional connections that do not get uniformly stimulated at repetitions. In later sections, I will use the term pattern extraction to describe the phenomenon of selecting the core synaptic pattern in the course of repetitions

Components of effective knowledge representation in active recall systems

All the principles of effective knowledge representation discussed in the following section come from years of experience in developing knowledge systems for my own use, as well as from a continuing opportunity to study the relationship between item difficulty and knowledge representation in knowledge systems developed by users of SuperMemo, many of whom seek professional advice before embarking on major knowledge system development, or equally often are eager to share their own experience and problems encountered while applying repetition spacing in all imaginable learning domains.

The following are the five thematic groups related to knowledge representation in self-instruction systems based on active recall:

In the five following sections I will address all the above thematic groups individually using examples from the aforementioned knowledge system on microeconomics

Sequencing items in the stepwise process of acquiring associative knowledge

The most important rule in sequencing items is inherently related to the learning process in general. The progression must go from basic concepts through foundations to more intricate and detailed issues. In all forms of learning, this principle derives from the need for comprehension, which obviously is greatly reduced when the student is thrown at a deep end at once. In application of repetition spacing with low forgetting index, this approach has also another important aspect. As in this form of learning, forgetting plays an insignificant role, the student is likely to experience the phenomenon of new pieces of knowledge nicely slotting in in the already established structure. This might affect the sequencing algorithm by moving it from truly basics-to-details approach to first-comprehended-first-memorized approach which dispenses with the need for full comprehension at the first contact with the newly learned material. This way the student will less likely experience the feeling of getting stuck because of his or her inability to comprehend some concept and, at the same time, reluctance to proceed any further out of fear of snowballing incomprehension. Thus, in the first pass of the material, the student will memorize only those components that have been comprehended, and hope that the slotting-in phenomenon will eliminate of comprehension gaps in the second pass.

The basics-to-details approach may be combined successfully with the most-applicable approach, in which the student delves more rapidly into details of those parts of the material which are most frequently referenced in other parts. This enhances the slotting-in phenomenon, which is one of the strongest motivational factors in learning, providing the student with the sense of accomplishment.

There is no cut-and-dried algorithm for optimally sequencing items then, however, in slightly more formal terms, the optimum sequencing might be defined as bicriterial optimization in which the following two factors are considered:

As an example let us consider the definition of the concept of production in economics. Production is commonly assumed to be synonymous with manufacturing. However, a more accurate and useful definition of production, from the economic analysis standpoint, is as any activity that creates value. The following item placed in the student’s memory is likely to profoundly affect the students interpretation of the concept:

Q: What is production?

A: any activity that creates value

As it will be shown later, it is always recommended to apply both the ability to associate the name with the concept, as well as the concept with the name; therefore, the following mirror item should also appear in the same database:

Q: What is the name of an activity that creates value (in economics)?

A: production

In the analysis of the aspects of production in economics, a more precise definition of production maybe useful for the sake of classifying its nature from the cybernetic standpoint. As the above definition provided an intuitive understanding and severed the link between production and manufacturing, the definition presented above, not quite consistent with the previous approach may have greater applicability in cases where the production process becomes the focus of more detailed analysis, esp. using economic models: "production is a series of activities by which resource inputs are transformed through a recipe and technological process into outputs of goods and services". Because of the fundamental nature of the concept, the very exact imprint of the above definition may be considered a valuable asset in building more advanced facets of knowledge of economics. The difficulty with basic concepts is that they are so basic that they cannot be asked for in yet more basic terms. Questions like "what is production?" demanding the by-heart recitation of the definition, as it will be shown later, entirely misses the point of building true comprehension (cf. simplicity of synaptic patterns and specificity of synaptic stimulation). Simplifying the answer to "series of activities" and following it with a number of items that define the "activities" is also inadmissible because of a number of adequate substitutes for "series of activities" like, for example, "any activity that creates value", and many more. Here, a very valuable, and often underappreciated tool of Cloze deletion comes handy. Consider the following wording of items:

Q: Production is a series of ... by which resource inputs are transformed through a recipe and technological process into outputs of goods and services

A: activities

similarly, Cloze deletion should generate items in which the following terms are missing in sequence: "resource inputs", "recipe", "technological process", "outputs", and "goods and services", so that to finally arrive at:

Q: Production is a series of activities by which resource inputs are transformed through a recipe and technological process into outputs of ...

A: services

The question arises if the aforementioned set of items will generate the desired effect which is the comprehension of the concept of production along the definition provided earlier. The experience shows that despite seemingly high degree of dissociation between the particular components of the learned concept, the presented items appear to produce a solid imprint in the students memory that not only provides a firm support for stable comprehension, but also makes it possible to effortlessly recite the entire definition of production. E-factors for such constructed items typically fall into the range from 2.0 to 2.8 depending on other elements affecting memorization with interference as the most prominent factor. Despite a larger number of items, this is a sure guarantee to produce less workload than in the case of cramming the entire definition in the answer with E-factor most likely to drop below 1.5.

For contrast, let as consider the following item that appeared to show a high degree of intractability because of the lack of respect for basics-to-details approach:

Q: What is discount rate?

A: interest rate charged by the FRS on loans to member banks

This item was frequently forgotten because it was not backed up by other items in the same database that would result in understanding the acronym FRS (Federal Reserve System), and consequently the concept of a member bank. This made the definition of discount rate leave little semantic connotation forcing the student to assume syntactic approach to memorization which is nothing else than mindless cramming with very poor retention prospects.

Techniques for minimizing the complexity of synaptic patterns as a key to keeping E-factors high

The most important principle of effective knowledge representation in systems based on active recall and repetition spacing is minimization of the complexity of synaptic patterns involved in storing memory engrams (Wozniak, 1990). This principle translates to keeping the content of question-answer items simple, specific, graphic, consistent, comprehensible and univocal. The main purpose of such an approach is to make sure that the spatiotemporal pattern of firing during the learning task is the same in each successive repetition. In other words, there should be minimum change to the synaptic pattern in the course of repetition as a result of pattern extraction. The entire concept of optimum repetition spacing is based on dealing with uniform pieces of information whose memory engrams are uniform and stable, and consequently can be treated as atomic entities. If the neuronal firing was to change its course over a number of repetition, a subset of synapses in the relevant synaptic pattern would not receive sufficient enhancement resulting in partial loss of the learned information.

Using examples from the microeconomics database, I will show all the distinguishable facets of minimization of synaptic patterns involved in representing individual pieces of information stored in the database.

Comprehension

It has been strongly pointed in the preceding section that the basics-to-details approach is, among other things, supposed to ensure the maximum level of comprehension. Here I will only note that comprehension is indeed related to the minimization of the complexity of synaptic patters that is the subject of this chapter. Nonsensical phrases or concepts involve a much greater number of neurons in the process of learning. Low-level electrical measurements showed that the neural activity is higher in cases of memorizing nonsensical words as opposed to natural words. Similarly, PET scans show that the brain activity of people exhibiting high IQ is much lower during performing a learning tasks that it is the case for low IQ students. Finally, it was shown that memory for people’s professions is more stable than the memory for their names. This was explained by psychologists as being related to the fact that well-established synaptic patters representing various professions are usually not matched by similar patterns that could be used easily used to represent names, esp. surnames.

Minimum information principle

Minimum information principle is the most obvious consequence of the approach based on minimum complexity of synaptic patterns. In order to keep the memory image of items simple, items should be simple themselves.

Let us consider the shortcomings of competitive markets such as unequal distribution of income, imposition of production costs on the public, development of socially undesirable products, product proliferation, etc. The minimum information principles says that the question "What are the shortcomings of competitive markets?" cannot be accepted because of the complexity of the answer. In such situations, solution comes from narrowing the focus of the question; the approach which often requires additional terminology and knowledge structuring, and is generally more demanding for the database developer. A typical question with a narrower focus might sound as follows:

Q: What problem with distribution of income appears on a competitive market?

A: income is concentrated in the hands of few

or,

Q: What is an example of imposing production costs on people who do not consume in competitive markets?

A: environmental pollution

The set of specific questions as those presented above produce a very high level of knowledge retention; however, the question arises if it is equivalent with the student’s being able to pinpoint the most important shortcomings of competitive markets. The experience shows that a number of items that would glue the above granules in a coherent entirety is necessary. This can conveniently be accomplished by means of Cloze deletions discussed later in the chapter. For example:

Q: The main shortcomings of competitive markets are:

- redistribution of income (the haves & have-nots)

- imposition of production costs (pollution)

- ... (illicit drugs)

- product proliferation (standardization issues)

A: socially undesirable products

The examples associated with particular shortcomings of competitive markets presented above serve as a vivid enhancement and comprehension booster, but their main function is to make it easier to track the missing clause. After all, as it will be shown later, enumerations are one of the trickiest obstacles to overcome in complying with the minimum information principle. The presented Cloze deletion serves as: (1) tool for mastering the terminology related to the discussed shortcomings, while the conceptual answer is indeed strongly suggested by examples accompanying the enumerations, (2) graphic skeleton for hooking up pieces of knowledge acquired by narrow-focus questions as presented earlier.

A yet more complex knowledge structure appears in the analysis of tax revenues in an attempt to plot the Laffer curve for European countries in the years 1975-1982. Upon the analysis, on the two ends of the spectrum, notable examples of two countries are worth considering: Sweden and Spain. The former, with the average tax rate of 49% showed 12% decline in tax revenue, while the latter with the average tax rate of 23% experienced a remarkable increase in tax revenues of 60%. Naturally, a single item cramming all the above facts has little chance of passing the minimum information criterion. Consider then the following items intended to ensure the student’s recall the facts related to tax rate vs tax revenue relationship:

Q: What was the average tax rate in Spain 1975-1982?

A: 23%

and

Q: What was the change in the tax revenue in Spain 1975-1982?

A: 60% increase

Unfortunately, similar questions asked for Sweden do not form a coherent memory image that would allow the student recall the entire collection of information pieces that make up the understanding of the relationship illustrated by the Laffer curve. Naturally, the understanding does not need examples. The theoretical implications of marginal tax revenue might be considered as a sufficient element of understanding; however, the usefulness of facts illustrating the theory has long been appreciated in education; I will therefore present for consideration an exemplary set of items acting as an associative glue for the discussed tax revenue case:

Q: In the years 1975-1982, the average tax rate and the tax revenue in Spain and Sweden were as follows:

Spain: ...% and 60% (respectively)

Sweden: 49% and -12% (respectively)

A: 23

and in a similar way:

Q: In the years 1975-1982, the average tax rate and the tax revenue in Spain and Sweden were as follows:

...: 23% and 60% (respectively)

Sweden: 49% and -12% (respectively)

A: Spain, etc., etc.

Items formulated in the above way appeared to produce very coherent memory engrams that showed above average retention rate despite the inherent intractability of numeric responses (as in the first of the two presented examples).

Narrowing by example

Narrowing by example is a very efficient way of making the question-related stimulus more specific and thus more successful in imprinting durable memories.

The concept of the price ceiling may be enhanced if a narrowing example of goods that might be subject to government imposed price ceiling is provided. Moreover, the example makes the definition of price ceiling more specific; hence the increased likelihood of diminished complexity of the synaptic pattern, and minimization of pattern extraction.

Q: What is the name of the price specified by the government above which goods cannot be sold (e.g. medications)?

A: price ceiling

In the example above, the phrases "(e.g. medications)" serves as the means of narrowing by example. Similarly, the illustration of competitive nature of pork versus beef helps narrowing by example in the definition of horizontal markets:

Q: What is the name of markets for products that can act as substitutes (e.g. pork and beef markets)?

A: horizontal markets

Note, that examples placed in the answer field will often act in the opposite way than it is the case above. The next section will discuss the various aspects of information redundancy in knowledge representation. In that context, the technique of extending by example will be presented.

Metaphoric approach

Reusing previously formed memories is the key to minimizing the complexity of synaptic patterns. This gives the preference to metaphoric rather than literal presentation of the drilled material.

The difference between demand and quantity demanded is that demand is described by the quantity-price curve, while quantity demanded is the value of demand for a given price level. The answer to the question "What is the difference between demand and quantity demanded?" might assume quite intricate wording if the above explanation was to be the way of grasping the learned difference. Instead, by extracting the essential nature of the difference in question, the following approach may be taken:

Q: What is the difference between demand and quantity demanded?

A: same as between a curve and a point

This approach is by far more effective in ensuring the student’s comprehension, and what is equally important, the resulting E-factor is, in most cases, very high. The recommended extension to the presented answer might be placed in parentheses with, for example, the following wording: "demand is described by the quantity-price curve, while quantity demanded refers to a single point on the demand curve".

Let us yet consider an example in which the metaphoric approach goes yet a step further by using nearly poetic language in order to describe economic concepts. In technology and innovation theories of profit, the creative effort of firms competing on the market can be translated into economic profit. In other words, technology and innovation are used to level monopolistic advantage of competitors, or to unbalance the conditions of perfect competition. By destroying the old, technology and innovation generates above normal profit. Here is a catchy item that is bound to show high E-factor value:

Q: What is the figurative statement that accurately reflects the working of technology and innovation in generating profit?

A: perennial gale of creative destruction

Note, that the later-discussed vivid and graphic elements in memorized items serve exactly the same purpose as the metaphoric approach: capitalizing on existing memories, or even on inborn neural structures, to form steady memory engrams.

Vivid approach

Formulating vivid, or even shock-evoking items, serves exactly the same purpose as metaphoric approach. The main difference here is that metaphoric approach capitalizes of existing declarative memories, while vivid approach makes use of the power of memories associated with circuits responsible for generating emotional impulsation. Here, derogatory terminology, humorous statements, reference to esthetics, taste, basic instincts, sex, etc. may serve as a powerful instrument enhancing memories. Additionally, vivid approach adds extra attractiveness to compliant databases, acting as a very desirable motivational factor.

In the world of business, the very typical approach to optimization of the firm’s performance is based on experience, intuition, guesswork and pure drive toward satisfying managers’ fancies. This drove a flock of business writers to mock the merits of economic analysis, and even to discourage potential graduates of business schools from enrolling. After all, the reasoning goes, there is no better school than running one’s own business oneself. The damaging impact of such an attitude has annoyed top economic brains more than once, and gave plenty of scope for derogatory statements, often tinted with emotional overtones. Instead of providing here the statistics referring to the career record of the graduates of Harvard School of Business, a single vivid statement of a noble scholar might serve as a sufficient incentive to study the theoretical aspects of running one’s own business:

Q: What was the opinion of Herbert Simon about the companies’ attitude toward profit maximization?

A: managers satisfice as they do not have wits to maximize

Additionally, adding a statement in parentheses indicating the Herbert Simon’s authority (Nobel prize in economics) might enhance the emotional overtones of the statement by providing a sharper contrast between the wisdom of a scholar and the close-mindedness of the gray mass of smug and self-satisfied business managers.

Graphic approach

The third technique based on capitalizing on previously established memories is the graphic approach. In previous sections, declarative memories and emotion memories have been employed. Graphic approach makes use of the powerful visual processing capability of the human brain. It has been discovered long ago that visual memories are by far more stable than verbal memories. Indeed, the core of mnemonic techniques makes use of the visual processing capabilities to enhance memory retention. In graphic approach, graphs, illustrations, photographs or video clips are used instead of verbal descriptions. As it will be shown later, textual representation of knowledge does not preclude graphic approach; nevertheless, it is usually a graphic figure that makes the most straightforward solution.

Enumeration techniques

One of the most fundamental principles in minimizing the complexity of synaptic patterns is to consequently avoid enumerations. Enumerations, especially with respect to sets as opposed to ordered lists, can be shown to result in an ambiguous impact on generating synaptic patterns. This can be observed even at the behavioral level, when at producing the response, the student is likely to stray with his or her thoughts, often producing the elements of the set in different order at each particular repetition. As mentioned earlier, variable synaptic stimulation at repetitions is likely to greatly reduce the effectiveness of memory consolidation; the net result being higher E-factor.

In can be shown that luxury products, novel products or products that have good substitutes show highly negative price elasticity of demand, while the demand for basic necessities or durable goods is rather inelastic. This important observation is an excellent example of knowledge that might most conveniently be represented as an enumeration: "What exemplary goods show highly negative price elasticity?". Naturally, unless combined with some mnemonic way of remembering, the enumeration is bound to cause persistent recall problems. The simplest workaround here is to formulate a collection of questions according to the following pattern:

Q: What is the price elasticity of demand for basic necessities?

A: inelastic demand

Q: What is the price elasticity of demand for novel products?

A: elastic demand

The main difference between the semantic memory image of the proposed collection of questions and the relevant enumeration is the student’s inability to recall all, or even a proportion of goods with elastic, or inelastic demand. However, the analytically more useful understanding of the factors that influence the value of price elasticity of demand is even better served. A partial solution to the noticed shortcoming, which does not bear the negatives of an enumeration might be:

Q: What exemplary goods show inelastic price demand (recall at least two)?

A: basic necessities, durable products, saturative goods, unique products, etc.

The remark in the parentheses plays an important role in ensuring that the student does not treat the above item as an enumeration, and that it clearly specifies the satisfactory degree of recall that can be used to decide when and when not to provide a passing grade.

Another approach can be taken in extending the definition of production discussed earlier and considering various kinds of resource inputs. These can most generally be: raw materials, labor, capital, land and managerial skills. An enumeration of exemplary raw materials might be reversed to a definition of raw materials through an enumerative example:

Q: What is the name of resource inputs such as coal, steel, water, etc.

A: raw materials

Finally, the most universal solution to enumerations are Cloze deletions. For example, instead of asking the student to recall different kinds of production such as unique product production, rigid mass production, flexible mass production and flow production, one might construct a series of Cloze deletion items in the following form:

Q: The types of production are:
- unique-product production (e.g. an office building)
- rigid mass production (e.g. old Ford's cars)
- ... (e.g. new GM cars)
- flow production (e.g. oil refinery)

A: flexible mass production

The above approach is universal and, in most cases, highly effective in dispensing with the enumeration problem.

Deletion and graphic deletion

Deletion is a simple technique that makes it possible to quickly generate collections of items derived from the same complex piece of knowledge, for example an intricate sentence. In an item based on Cloze deletion, the question presents a coherent piece of knowledge with a single element missing (standardly replaced with three dots), while the answer provides the missing element. In the previous sections, a number of deletions have been presented in the context of item sequencing, minimum information principle and enumeration techniques. Here, I would only like to focus on graphic deletions and Cloze deletions that may capitalize on their graphic aspects.

Graphic deletions differ from previously presented deletions in this way that instead of textual elements, pieces of graphs are deleted or obstructed, and the answer might provide the missing piece or its name. Because of basing on visual processing capability of the cerebral cortex, graphic deletions are a powerful tool for representing knowledge with a view to minimizing pattern extraction in the course of repetitions.

Deletion does not have to be graphic to make use of visual processing powers of the brain. The mere spatial distribution of particular textual components may evoke visual images that will increase the efficacy of recall. If enumerative deletions do not change the sequence in which enumerated elements appear in the question, their spatial location will strongly be imprinted in the students memory despite little emphasis on location at the recall time. This makes it easier to graphically visualize the enumeration. As a consequence, it is not unusual that the student will be able to recall the entire enumeration in spite of the fact that indeed neither of the items in the deletion group requires the knowledge of the enumeration itself to qualify for a passing grade at repetitions.

Here again is an example of an item based on Cloze deletion. The important thing to notice is the clear visual image of the three-part enumerative structure evoked in the process of learning.

Q: The options for a higher cost company in competition with a more cost-effective rival:
- ... (big losses expected)
- collusion (rather illegal)
- improvement (of the product or cost structure)

A: price war

Naturally, at least two other items of the same sort will appear in a well-structured database; the missing component being collusion and product of cost-efficiency improvement. Additionally, elements of the declarative part preceding the enumeration might be made missing. As a result, several items will work on enhancing the visual component of the textual structure; the net result being better recall of the elements and the entire piece of knowledge.

Dismembering complex concepts

Until now, only relatively simple elements of the learned knowledge have been considered; however, some concepts may be best grasped by simultaneous understanding of a number of subconcepts put together in a tightly interlinked mesh. This will often include control systems, mathematical techniques, complex theoretical models, etc. The main striking difference between such complex concepts and the examples presented before is that the single meaningful unit of knowledge can be expressed in a single sentence or passage in the latter case, while the complex concepts might span several paragraphs, none of which might be taken separately as a meaningful whole. It is not true that the complexity of such concepts cannot be unraveled. The main reason for their existence is not any inherent property, but lack of, or no need for specialized terminology that might serve to separate the smaller units. Dealing with such concepts is particularly difficult, and requires special skills from the knowledge system developer. Very often, the ultimate solution comes from introducing new terminology that is suitable for describing all subcomponents separately.

As an example of a complex concept in economics, and the means of dismembering it into manageable pieces of knowledge compliant with minimum information principle, I will consider the determination of the utility-maximizing combination of products subject to an income constraint.
Let Pa, Pb, Pc, ..., Pn be the prices of products Xa, Xb, Xc, ..., Xn, I be a consumer’s money income, and TU=f(Xa,Xb,Xc,...,Xn) be the consumer’s utility function for n products. The total utility function is supposed to be maximized subject to the income constraint in the form:

I=Pa*Xa+Pb*Xb+...+Pn*Xn

A LaGrange multiplier l is introduced to combine the total utility function with the income constraint in order to produce the function Z subject to further analysis:

Z=f(Xa,Xb,...,Xn)+ l*(I-Pa*Xa-...-Pn*Xn)

The partial derivatives of Z are found for each variable and equated to zero:

  • Z/Xa=TU/Xa-l*Pa=0, etc.
  • Z/l=I-Pa*Xa-...-Pn*Xn=0

These equations can be solved to determine the utility-maximizing purchase levels for Xa, Xb, ..., Xn. Soon we arrive at:
(TU/Xa)/Pa=(TU/Xb)/Pb=...=(TU/Xn)/Pn
that is equivalent to MUxa/Pa=MUxb/Pb=...=MUxn/Pn, where MUxi is the marginal utility of product Xi. The above equation expresses the condition for maximum utility in the purchase of a group of products.
Here is how the above derivation could be expressed in an active recall system compliant with the principle of minimum information:

Q: What is the formula for total utility function in maximum utility analysis?

A: TU=f(Xa,Xb,...,Xn)

Q: What is the formula for income constraint in maximum utility analysis?

A: I=Pa*Xa+Pb*Xb+...+Pn*Xn

Q: What is the name of the l factor used in maximum utility analysis?

A: LaGrange multiplier

Q: How is the total utility function and the income constraint combined in maximum utility analysis?

A: Z=TU+l*(Pa*Xa+...+Pn*Xn)

Were it not for the possible misunderstanding, the above expression might yet be shortened to Z=TU+l*I. The more complex formula used above has been opted for only because of the derivation step that follows, for which case the income per se is of no use.

Q: How is the function Z used to find the optimum combination of Xa, Xb, ..., Xn in maximum utility analysis?

A: partial differentiation and equation with zero

Optionally, the results of the derivation for Xa and l might be added here for easier recall of particular steps of computation and their meaning.

Q: What is the final conclusion coming from finding the combination of Xa, Xb, ..., Xn that maximizes function Z in maximum utility analysis?

A: MUxa/Pa=...=MUxn/Pn

The above set of items is used only as an introductory illustration and should yet be extended to comply with redundancy principles presented in later sections. It is clearly visible that good terminology is a key to effective dismemberment of complex concepts into simple question-answer items. The most visible terminological shortcomings of the passage presented above is lack of an accurate term to describe the function Z, which, taken out of context, is absolutely meaningless. Secondly, the short and catchy term of maximum utility analysis has been concocted only for the purpose of not having to use the much longer name of the determination of the utility-maximizing combination of products subject to an income constraint

Mnemonic techniques

Mnemonic techniques go one step further than the graphic approach in this sense that they use artificially redundant graphic images to represent unique or nonsensical information. One of the two fundamental mnemonic techniques are mind maps and peg lists. A mind map is a graph that in a vivid form represents the structure of semantic connections between particular components of the learned knowledge. A graphic model of market economy might be an example of a mind map; however, mind maps are applicable also to all pieces of semantically coherent knowledge, independent of their usual form of representation in standard textbooks. Thus, the computation of the combination of products that maximizes the customer’s total utility (see previous section) might also be presented in a graphic form. A flow chart is the most impelling proposition, though any other form of a not necessarily directed graph might do. An interesting variety of a mind map is a graph that is mapped on the image of a familiar object, e.g. one’s own apartment. Retrieving particular pieces of such a graph from memory is particularly easy, though the solution is not always universal because of the fact that each student would rather use mapping familiar for him or herself. The most often applied universal mapping is the one that pegs particular nodes of the mind map to the parts of the human body. The main shortcoming of the presented approach is strong interference between multiple mind maps pegged to the same object.
Another popular mnemonic technique is peg lists. A peg list is a sequence of well visualized objects associated with cardinal numbers. For sophisticated applications, peg lists usually consist of 101 objects pegged to numbers from 0 to 100. The main application of peg lists is in remembering numbers and ordered enumerations. A 101-element peg list can be used to represent all numbers as sequential visual scenes composed of peg list equivalents of two-figure components of the remembered number. For example, assume that a phone number 867045 is supposed to be memorized by means of a peg list. Assume that the following images are associated with the two-figure components of the number: 86 - car (first car was built by Carl Benz in 1886), 70 - phone (Graham Bell invented the phone in 1870), and 45 - bomb (the date of Hiroshima bombing). If we imagine a scene in which we drive a car, pick-up a mobile phone and spark off a great fireball by activating the ringer, then we have effectively mapped an otherwise nonsensic phone number onto an easily retrievable graphic scene (the mapping being effected through the peg list).

To illustrate the phenomenon of increasing returns to scale, and the incredible competitive advantage the Ford Motor Company has gained in the early 1900s over its competitors though specialization of labor based on semi-automated assembly lines, the student might wish to note that in 1914 the FMC produced 270,000 cars with 13,000 employees; while the other 299 American auto companies at the same time, with 66,000 employees produced just 290,000 cars. The example posses a serious dilemma to a database developer. Each of the number quoted makes up useless garbage knowledge. However, taken together, the figure combine into a vivid and compelling illustration of increasing returns to scale and their importance in running any kind of business. Demanding from the student the understanding of increasing returns to scale deprives the example from its strong emotional overtones, as the student might identify him or herself with Henry Ford’s business cunning. Depriving the example from numbers takes a great deal of its vividness. The two proposed solutions are: (1) limit the question to an estimated figure that shows the FMC lead in the market, and (2) use Cloze deletion to dismember the above sentence, and use mnemonics to memorize the involved numbers. The first approach might look as follows:

Q: What was the share of the American automobile market commanded by the Ford Motor Company in 1914?

A: Close to 50%

or using Cloze deletion and mnemonic techniques:

Q: In 1914 the Ford Motor Company produced 270,000 cars with 13,000 employees; the other ... American auto companies, with 66,000 employees produced just 290,000 cars.

A: 299 (Ford turns on a light switch to see how many competitors he has got, and ... only two cats spring up turning their tails)

The seemingly flippant comments in the parentheses above are part and parcel of mnemonic representation. In the example above, an eleven-member peg list has been used with the number two represented by a light switch (the switch has two states: on and off), and nine represented by a cat ("cat has nine lives").

As the analysis of intractable items in numerous databases show that numbers take the lead in making items indigestible to human memory, the use of numbers in databases of all sort should be limited to the absolute minimum. As the discussed database on microeconomics was notably sparse in numbers (mathematical formulas do not count here), the above example was a notable exception, and, perhaps for that reason, did not cause any serious recall problems. However, had there been more such numerically saturated cases, the issue could have started being a problem.

Item univocality and inter-item interference

Univocality of items is not as much about minimizing the complexity of the synaptic patterns as it is about making sure that disjoint patterns are used for different items. Similar wording or even similar associations evoked by two separate items results in inter-item interference, which very often leads to confusion, providing wrong answers in reference to well-remembered pieces of knowledge, insufficient neural stimulation, and lack of uniform memory consolidation over the synapses involved in both the interfering and the interfered with synaptic pattern.

A very typical interference problem is the terminological ambiguity. For example, there is nothing wrong with the question "What is the formula for marginal rate of substitution?" as long as the substitution of two competing products is concerned. However, as soon as we move to the isoquant analysis of the production process, the marginal rate of substitution of capital for labor starts interfering with the until now simple picture. Indeed, in the latter case, the correct term to use is the marginal rate of technical substitution; however, this terminological nuance does not help much to eliminate the interference problem. A very simple solution to the above interference problem is to provide strong context clues in the question. For example:

Q: What is the formula for marginal rate of substitution of products X and Y?

A: dX/dY

Q: What is the formula for marginal rate of technical substitution of capital and labor?

A: dC/dL

Though the solutions to the problem of interference usually appear to be very simple, the mere process of locating potentially interfering item poses a formidable challenge to a learning material developer. Indeed, there is only one true and tried method of eliminating interfering items: memorizing the entire material. Only the neural network of the human brain can on-the-fly spot the problematic similarities. This clearly illustrates the fact that practically no learning material designed for active recall in repetition spacing should be designed in detachment from the natural learning process. This naturally increases the development costs manifold.

Let us now consider a case of strong semantic inter-item interference. The following items all deal with the problem of diseconomies of scale; however, no explicit interference problem strikes the eye at first:

Q: What is the frequently quoted argument for the U-shaped cost function?

A: most companies work at about 90% of their maximum capacity

Q: Why does the theory claim that every firm must reach a point of constant returns to scale with increasing output?

A: by striving to push the production to the limits, the company must at some point decrease its cost efficiency (overloading people, machinery, facilities, etc.)

Q: Why corporations may encounter problems when growing beyond a certain point (cf. River Rouge plant)?

A: because of managerial problems

Q: What is the main factor of diseconomies of scale in the US economy?

A: trade union activity

Upon a closer look, it appears that semantically, all the above items ask the same question "What are the reasons of diseconomies of scale?", while each of the item provides a different answer. Naturally, this is a trouble in the making for the student. It want take long before he or she starts confusing the River Rouge case with trade union activity problems, or attribute the U-shaped cost curve to dimension-related managerial problems. In a well-designed, interference free learning material, there is little choice but to use some of enumeration techniques to list the most important factors contributing to decreasing returns to scale.

As I tried to demonstrate, the authors of learning material have little choice but to memorize their own material before making it available to a wider group of students. Both the terminological and semantic interference may reduce the effectiveness of working with self-instruction systems based on active recall.

Planned redundancy as a way to cross-strengthening synaptic patterns

In this subchapter I will discuss techniques that, in a sense, go against the approach based on minimum complexity of synaptic patterns. Namely, I will show the importance of redundancy of knowledge representation in effective recall of information. The paradoxical contradiction between the previous and the presented approach can quickly be resolved if we notice that the redundancy is not understood here as adding extra components to otherwise minimally intricate synaptic patterns. The function of redundancy is here exclusively to promote the establishment of additional synaptic patterns serving as emergency access routes to the remembered knowledge. Redundant items will by no means duplicate their content in the database, at least not in the syntactic terms. This, first of all, would go against the principles of the repetition spacing algorithm, which assumes the uniqueness of items as one of its fundamental premises. However, the same semantic contents might be expressed using different means for the sake of providing the pattern-matching neural network of the brain with an opportunity to derive the semantic common denominator (as, for instance, in items that use multiple narrowing by example). The derivation of the common denominator will naturally proceed through the mechanism of pattern extraction. The redundancy will generally comprise the following elements:

The main function of redundancy is not to make items easy to remember, but to make sure that forgetting an item does not affect the entire associative structure of the knowledge graph. Forgetting, as it should be stressed here, makes an inherent part of repetition spacing algorithms, and cannot be by any means eliminated. The ideal model of 100% retention is for biological reasons unfeasible. Redundancy is supposed to minimize the possible effects of forgetting on the performance of the learned skill.

Passive and active approach

The simplest illustration of redundancy via passivization is in the case of learning new terminology. The basic idea is to construct items in such a way that the definition of the concept and its name are placed one’s as a question and once as an item. If the definition appears in the question, the brain makes an association between the concept and its name. If the name appears in the question, the brain learns how to recognize concepts by name. Although, very often learning the name of a defined concept is sufficient to passively recognize the concept by name, it is not always so; hence the importance of the redundant approach. Additionally, even if one of the item gets caught in the forgotten pool, the other may serve as the way to restore the forgotten memory. In other words, presenting concepts both in passive and active form serves on one hand as an extension of the memorized pattern and as a protection against incidental forgetting on the other.

As an example of passive and active approach consider the following item:

Q: What is the name of the curve determined by the quantities of two (or more) products in combinations that produce the same total utility?

A: indifference curve

Q: What is an indifference curve?

A: curve determined by the quantities of products that produce the same utility

It is worth noting that in the second item, the answer has been simplified to the maximum possible extent to reduce its expected E-factor. Any possible inconsistencies resulting from such as simplification should rather be resolved by adding new items on the subject rather than making the answer more intricate.

Support for derivation, reasoning and intelligence

The ageless dilemma of elements of nature vs. nurture in education has from the very beginning been bound to crop up in this chapter. I will try to show that elements of what is commonly understood as intelligence can be developed in the process of learning based on active recall and repetition spacing. There are two definitions of intelligence that are in common use, often without distinguishing one from the other. On one hand, intelligence can be understood as the brain’s ability to process information. On the other, the potential to develop such an ability is often used interchangeably with the ability itself. When we speak of somebody "he will solve this problem quickly; he is a very intelligent person" we use more of the first interpretation of intelligence. However, when we say "this student will do a great deal in his life; he is very intelligent" we employ more of the second interpretation. It is easy to notice that a great deal of inborn characteristics will influence intelligence in the sense of the potential to developing good information processing capabilities. Apart from the number of neurons and neuronal connections, development of the glia and other elements that give the nervous tissue more scope for plasticity, such personality characteristics as mental and emotional stability, high levels of serotonin and dopamine, etc. may play equally important role in developing intelligence. If we, however, consider intelligence as the ability to process information, most of it will, as I will try to show, be developed in the course of education. What makes a bright mathematician is not just mathematical knowledge, not inborn talent, but the ability to associate various components of his or her knowledge of problem solving in mathematics. Consequently, properly represented knowledge of mathematical concepts, and more importantly, of mathematical reasoning will distinguish the nimble brain of a problem solver from the average mortal. The core knowledge of intelligent thinking, in mathematics and beyond, is the rules of mathematical derivation in the most abstract and universally applicable form. Those rules can be applied in a myriad of daily situations. This universal applicability in problem solving makes the basis of what others consider an intelligent person. If properly formulated and represented for learning, these rules can be memorized in a standard way; in other words, memorization can be a way toward intelligence!

In this section, I will show several examples in which reasoning steps are entangled in the knowledge structure represented as question-answer items used in learning based on repetition spacing. A typical situation is when we have the general definition of a problem (i.e. not a particular problem instance), and we initially memorize the solution to the problem. For example, we might have the definition of the maximum utility problem (see Dismembering complex concepts) combined with its solution of equal marginal utility for particular products. In case of significant importance of the particular problem-solution pair, adding particular reasoning steps (here partial differentiation and equating the results with zero) might on one hand provide derivation rules that contribute to the ability to solve similar problems and on the other provide a dose of redundancy whose role in sustaining memories has been highlighted earlier. It can be easily shown that eliminating the derivation steps will free the student’s memory from considering the steps of reasoning during repetitions. Additionally, requiring the student to solve the problem on his or her own each time the repetition takes place can be equated with enumerative learning which as I earlier tried to demonstrate, stands against the rule of minimizing the complexity of synaptic patterns, which is central to effective learning! In other words, apart from solving particular problem instances as a form of repetition, I see no reasonable alternative to pure memorization of derivation steps as the best means of boosting the student’s problem solving capability (i.e. intelligence in the first of the mentioned interpretations). The general rule then is: whenever possible and sensible, memorize the derivation steps of a particular solution to a general problem.

Let us consider the concept of perfect substitutes as a simple example of providing knowledge extensions that should otherwise be deducible from previously learned facts and rules.

Q: What is the name of products that are characterized by the same constant marginal utility?

A: perfect substitutes

Q: What is the shape of the indifference curve for perfect substitutes?

A: linear

Q: What is the name of two products whose indifference curve is linear?

A: perfect substitutes

The fact that the utility function of two products is the same, makes it possible to conclude that their indifference curve must be linear. However, the mere knowledge of the concepts definition will not by any means reinforce the understanding of this fact. In other words, the student may need substantial time to conclude the shape of the indifference curve. Having the fact memorized explicitly, not only smoothes up the derivation pathway from the definition of perfect substitutes to the shape of their indifference curve, but also serves as a strengthener of a more general and abstract rule which says that the derivative of the sum of functions with respect to a variable is equal to the sum of the derivatives. Naturally, the degree of strengthening depends on the student’s wish to use reasoning to derive answers rather than pure syntactic memory. In addition, the link between perfect substitutes and the linear indifference curve is reinforced through the reversal of the question and answer fields.
Having memorized the above facts the student shows a higher degree of understanding of the concept of perfect substitutes, as well as a quicker response time in derivation tasks based on mathematically akin concepts. It is important that the derivation steps are short enough to comply with the principle of minimum complexity of synaptic patterns. Longer derivations might cause insufficient memory stimulation, and despite their being excellent problem solving drills, the value of their E-factor might turn them into intractable elements of the database that result in disillusionment and lack of enthusiasm on the part of the student.

Similar situation we can see in the case of the definition of complementary products, which can be amplified by the concept of cross-elasticity:

Q: What is the name of products X and Y such that increasing purchases of X increases the purchases of Y?

A: complementary products

Q: What is the formula for cross-elasticity of products X and Y?

A: dX/dPy*(Py/X) (where Py is price of X)

Q: What is the name of products with negative cross-elasticity?

A: complementary products

Here, the third item serves as a memory strengthener for both complementary products and cross-elasticity. Additionally it serves as a derivation step that applies the abstract rule of the sign of a derivative.

Finally, I would like to show a trivial derivation step that may indeed be crucial for retrieving memories. Consider the following item:

Q: How can total revenue be computed from the demand curve at a given point?

A: price*quantity

Does it makes sense then to enhance the above item with one that is its direct and trivial consequence?

Q: Can total revenue be computed from the demand curve?

A: yes

Note, that the latter item sets thinking in terms of feasibility, not the procedure, algorithm or implementation. It can be shown, that a student who knows the procedure needed to execute to obtain the result, will not even attempt the execution because of lack of the feasibility conviction! In other words, one may be tempted not to ever try arriving at the solution. Thinking in terms of feasibility as opposed to thinking in terms of the procedure, give the reasoning two different contexts that might yield two different outputs. Such minor memory associations as presented in the above item, taken together, contribute to what is generally described as the problem solving ability.

Optional reasoning clues, mnemonic clues, context and examples

The fact that items should comply with the minimum information principle does not mean that they cannot be in any way redundant in itself. It is only important to make sure that the compulsory content that is subject to the repetition does not contain redundant elements. Apart from that, the item itself may contain a great deal of accessory material that might be useful in learning. This can include context clues and explanations, reasoning clues, mnemonic clues, illustrative examples, or even hypertext links, etc. Each time it must only be clearly specified that the redundant contents is not compulsory and in any way needed for scoring the passing grade.

It is well known that the total utility derived by the consumer from a number of goods is not the arithmetic sum of the particular utilities. This property derives from the fact that products may enhance or suppress each other’s utility. This fact can be comprised in an item formulated as follows:

Q: Why isn’t the total utility function a sum of utilities of particular products?

A: because products may mutually enhance or suppress their utility

As I tried to show in the preceding section, a simple derivation step might enhance the student’s deductive ability with reference to the above piece of knowledge. This could be accomplished by a simple question like "Is the utility function a sum of utilities of particular products?". However, it appears to be useful to provide some redundancy to the answer element (which in this case is simply "no"):

Q: Is the total utility function a sum of utilities of particular products?

A: no (because products may mutually enhance or suppress their utility)

Seemingly, this item became a sister copy of the one mentioned earlier. However, the compulsory semantic connection to be recalled in order to score a passing grade is different in these two cases. Again one item refers to the procedure, the other to feasibility. The explanatory part placed in parentheses is by no means needed to pass the repetition and serves exclusively as a memory strengthener, reasoning clue and reference note. The student may opt not to read the explanation at repetitions at all. However, if her or she notices that his or her response became automatic rather than semantic, the reasoning clue may serve to restore the right context and ground for the answer.

Earlier, I presented an example of an item that used a mnemonic peg list as a support in remembering numerical responses. Mnemonic clues placed in parentheses may be the best way of remembering numbers; however, in some cases the number itself sticks easily to memory and the mnemonic part becomes unnecessary. It must be remembered, however, that in order to comply with the principle of uniform synaptic stimulation at repetition, the student should clearly set his or her mind or either using or ignoring the mnemonic clue.

Context clues and notational conventions may help making sure that the established memory link does not become meaningless in time, or worst of all, associated with wrong context.

The concept of marginal revenue expressed in mathematical term is much easier to comprehend and retain in memory that its verbal equivalent. Here, however, it is strongly recommended that all symbols used in the equation be explained in an explanatory note, which does not take part in the learning process itself, and is used only as a verification means in case the used symbols started losing their meaning with the increase in the length of the inter-repetition interval.

Q: What is the formula for marginal revenue?

A: MR=dTR/dQ (dTR - change in total revenue, dQ - change in quantity sold)

Finally, for highly associative knowledge, additional explanatory links, or even hypertext links, might help keeping the overall knowledge structure in place.

For example, understanding the meaning of the Laffer curve does not necessarily entail the ability to recall its shape. Naturally, all independent pieces of knowledge should be placed in independent items; however, it is also reasonable to make sure that each time the Laffer curve crops up, its shape appears in the student’s imagination (even without the need to resort to graphics):

Q: What does the Laffer curve express?

A: dependence of tax revenue on tax rate (minimum revenue for very high and very low tax rates)

Complexity of wording vs. comprehension

Simplicity of item wording does not necessarily equate with the minimum information principle. The latter puts emphasis on the minimum complexity of the knowledge as it is represented in student’s memory. There is no imperative for items to be complex in their textual or graphic representation in the database itself. After all, some very simple concept may require quite a great deal of text to describe them verbally. However, it can be shown that excessive complexity of items as represented in the database may negatively affect the process of learning. The main problem is in miscomprehension and confusing different items through reading that naturally tends to be quick when the student is confronted with a massive number of repetitions per learning session. Very often, even the mere selection of words may affect the comprehension. As previously, also in such cases, the best remedy is the database author’s own work on memorizing the database in question and eliminating possible snags on a one by one basis. Such minor elements as moving a single phrase to a separate line may greatly help keeping E-factors high.

Additional functionality encapsulated in items

Finally, I would like to note that a database used for learning may, and usually is employed also for other purposes than acquiring knowledge. These may include archiving and retrieval purposes, literature sourcing for publication references, date stamping for facts that quickly change in time, individual mnemonic clues (i.e. clues related strictly to a particular student’s knowledge or life), domain labels, ordinal numbers for sorting databases, and many more.

Government budgets change almost as often as governments themselves. Therefore it seems necessary to provide date stamping in the following question:

Q: What proportion of Japan GDP is spent on R&D (1990)?

A: 2.9%

The date in the parentheses has no bearing on the learning process as long as there is only one item in the database that pertains to Japanese R&D budget. However, it may appear useful in cases when the student deems necessary to update the figure after some time when it becomes exceedingly outdated.

A more extensive stamping can be seen in the example that follows.
There is no rule as to what proportion of sales ends up as profit after taxes. However, global economic analysis makes it possible to establish the approximate figure for an average company. The main problem here is that the final value may strongly depend on the current economic situation of a particular country, as well as on the methods applied in the analysis. In the example below, the item provides the source of information that was used to estimate the net margin of profit.

Q: What is the average proportion of sales that ends up as profit after taxes (based on accounting definition of profit)?

A: 4-6% (The Economic Report of the President, February 1988, pp. 352-353)

Obviously, only the figure of profit after taxes is subject to learning. The source specification is provided additionally for archiving purposes or for publication purposes (e.g. in case the student wished to refer to the figure in one of his or her publications).


Summary of knowledge representation issues in learning


See also: