Planned redundancy as a way to cross-strengthening synaptic patterns
(from P.A.Wozniak, Economics of Learning)
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 knowledge system, 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:
- using both active and passive recall
- providing accessory reasoning derivation steps
- providing optional reasoning context
- providing multiple representation for semantically homologous items
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.
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 A-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.
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 A-factor might turn them into intractable elements of the knowledge system 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)