Step Size

In simple cases of stimulus-response conditioning—in which a single response is to be conditioned to a single stimulus—the step size of learning, or increment of graduation in attainment, is defined by (a) the number of differentiable stimulus elements present during each trial; (b) the variation in stimulus elements from one trial to the next; and (c) the prominence or significance of the stimulus to which a response is intended to be conditioned, relative to the other stimulus elements that are also present. The effect of step size on time required for learning, in cases of simple conditioning, is most obvious during conditioning for discrimination or generalization, in which too great a variation in stimulus attributes from one trial to the next fail to evoke the desired response. For more complex learning in which multiple associations or bonds are formed—e.g., (a) Thorndike’s example of improvement in addition of one place numbers, (b) his example of improvement in typewriting by sight method (Thorndike, 1914a, p. 228), or (c) Guthrie’s explanation of the learning of acts, which he said are made up of many individual movements (Guthrie, 1942, p. 36)—step size takes on greater dimensionality in accounting not only for variation in elements of the stimulus condition but also the number of bonds or associations to be formed and the ease of formation of each bond (Thorndike, 1914a, p. 229). In simple conditioning a single stimulus is conditioned to a single response. In complex conditioning, multiple stimulus-response pairs (i.e., bonds or associations) are conditioned, and step size is determined both by variation in the elements of each stimulus condition as well as the total number of stimulus-response pairs to be conditioned.

Variation in the stimulus condition may be intentional, but may also be unavoidable. In his fourth postulate Hull assumed a fixed incremental increase of habit strength as a summative result of each pairing of the stimulus and response (Hull, 1943, p. 178) even though he recognized that the conditioned stimulus is “nearly always a very complicated compound” which is “almost never repeated exactly” (Hull, 1942, p. 73). Skinner found that by reinforcing small increments of, or even only approximations to, the goal behavior he was able to condition complex behaviors that did not naturally occur in the organisms repertoire—i.e., teaching a pigeon to kick a ball (Skinner, 1961f, p. 132). In Este’s sampling theory, the step size of learning was defined by the number of stimulus elements in each trial that were not yet conditioned to the desired response. Too large of a step size would result in elements previously conditioned to the desired response to become conditioned to an alternative response, thereby undoing previous learning. “On each occurrence of a response, R1, all new elements (i.e., elements not already conditioned to R1) in the momentary effective sample of stimulus elements, s, become conditioned to R1” and “the conditioning of a stimulus event to one R automatically involves the breaking of any pre-existing conditional relationships with other R‘s. (W. K. Estes, 1950, p. 97).

Step size, as a principle of learning, is inferable from behavioral learning theory through the relationship between variations in the time required for simple conditioning to occur combined with noted cross-trial variations in the stimulus condition. In this context, step size is primarily the number of critical elements introduced or varied from one trial to the next. In discrimination learning or conditioning for generalization step size is the size of deviation in the stimulus condition from what has previously been conditioned. For more complex conditioning, as in shaping or chaining, step size is the increment of approximation towards the goal behavior (shaping) or the length of a sequence of behaviors that will be joined (chaining).

Even stronger evidence suggesting the reality of this principle comes from cognitive learning theory, in which it was addressed directly. Ebbinghaus, for example, investigated the “rapidity of learning series of syllables as a function of their length” (Ebbinghaus, 1913, pp. 46-51), and concluded that “in the cases examined, the number of repetitions necessary for the memorization of series in which the number of syllables progressively increased, itself increases with extraordinary rapidity with the increase in number of the syllables” (Ebbinghaus, 1913, p. 48). From this account we add amount of new content to be learned to our understanding of step size. Atkinson and Shiffrin (1968) explained the general phenomenon reported by Ebbinghaus in terms of (a) a process of transferring information from short term memory into long term memory, and (b) the concept of a limited rehearsal buffer. They stated that, “throughout the period that information resides in the short-term store, transfer takes” (p. 27), however, the amount of information that can be held in short term store is limited and decays rapidly. By means of a limited capacity rehearsal buffer (p. 36) information is kept active in short term memory through maintenance rehearsal and encoded into long term memory through elaborative rehearsal. Encoding processes are assumed to suffer if an attempt is made to hold too much information in the rehearsal buffer:

Presumably a buffer is set up and used in an attempt to maximize performance in certain situations. In setting up a maximal-sized buffer, however, the subject is devoting all his effort to rehearsal and not engaging in other processes such as coding and hypothesis testing. In situations, therefore, where coding, long-term search, hypothesis testing, and other mechanisms appreciably improve performance, it is likely that a trade-off will occur in which the buffer size will he reduced and rehearsal may even become somewhat random while coding and other strategies increase. (R. C. Atkinson & Shiffrin, 1968, p. 37)

Another cognitive theory perspective on step size was described by Ausubel et al. (1978) in terms of the complexity of the task, noting that “some tasks are so complex that they cannot be learned directly;” that “the learner must first be trained on a simplified version of the task and then transfer this training to an attempt at mastering the task itself” (p. 200). He recommended that “the content and step size of subsequent practice trials [should] be differentially adjusted in terms of the individual learner’s success or failure and type of error on preceding learning tasks or test items” (p. 339).

One final perspective from cognitive learning theory is different from the foregoing views in the following way. Each has suggested a need to restrict the increment of change to a manageable size. While this is also true of schema theory as far as learning through accretion and tuning go, it has contrastingly been suggested that there is a need for a critical amount of information to be amassed before the third type of learning, restructuring of one’s schema, will take place (Rumelhart & Norman, 1976, p. 4). Thus, on the one hand smaller increments of learning are more easily and more quickly achieved than larger ones, but, at least in some cases, the impetus for change must be sizable enough so as to cause its occurrence.

Similar to the requirement of a minimal critical mass for change in schema restructuring is the view in constructivist learning theory that curriculum should be structured around primary concepts in a holistic way, rather than separated into parts as is typically the case in traditional education (Brooks & Brooks, 1993). This suggests a measure of step size in terms of concept independence, where the optimal step size is one in which the information presented is complete enough to stand as a meaningful whole that presents a framework to which detail can be added.

Because Piaget saw development as a very smooth and “continuous progression from spontaneous movements and reflexes to acquired habits and from the latter to intelligence” (Piaget & Inhelder, 1969, p. 5) one can suppose that he might have considered the step size of learning to be quite small. Bruner spoke more clearly on this aspect of learning and described a measure of step size in terms of the number of component operations that make up a complete act. He took the position (J. S. Bruner, 1964, p. 2) that “there are very few single or simple adult acts that cannot be performed by a young child,” that “any more highly skilled activity can be decomposed into simpler components, each of which can be carried out by a less skilled operator,” and that “what higher skills require is that the component operations be combined.” Through integration “acts are organized into higher-order ensembles, making possible the use of larger and larger units of information for the solution of particular problems” (J. S. Bruner, 1964, p. 1).

Vygotsky describes step size in terms of incremental attainment through repeated exposure in a spiral pattern (Vygotsky, 1978, p. 56) and the constantly changing zone of proximal development (p. 86) which is created as learning marches out ahead “in advance of development” (p. 86). Bandura clearly demonstrated the principle of step size as he used response induction aids to assist clinical patients overcome their fear of snakes (Bandura, Jeffery, & Wright, 1974)—modifying the step size as needed so as not to go too fast, and introducing induction aids such as gloves before direct skin contact. Bandura also explained that the amount of observational learning that will take place depends on the availability of component skills:

The amount of observational learning that will be exhibited behaviorally partly depends on the availability of component skills. Learners who possess the constituent elements can easily integrate them to produce the new patterns; but if some of these response components are lacking, behavioral reproduction will be faulty. When deficits exists, then the basic subskills required for complex performances must first be developed by modeling and practice. (Bandura, 1977b, pp. 27-28)

From this we see that the size of the step being taken depends on the component skills that are already available. A learner possessing all of the component skills for a larger act need only integrate and coordinate them, while a learning lacking the component skills must also acquire them. Thus, the latter represents a case of larger step size than the first. Bandura also described a process of attenuated scaffolding to produce lasting changes in self-efficacy:

Results of recent studies support the thesis that generalized, lasting changes in self-efficacy and behavior can best be achieved by participant methods using powerful induction procedures initially to develop capabilities, then removing external aids to verify personal efficacy, then finally using self-directed mastery to strengthen and generalize expectations of personal efficacy (Bandura et al., 1975). (Bandura, 1977a, p. 202)

In situated learning step size is controlled through the gradual increase in responsibility (Lave & Wenger, 1991, p. 69). In activity theory, as has already been mentioned, large-scale cycles of learning involve numerous smaller cycles of learning actions  (Engestrom, 2010, p. 11). In cognitive apprenticeship, the step size of learning is managed through scaffolding and fading:

Scaffolding refers to the supports the teacher provides to help the student carry out the task. These supports can take either the forms of suggestions or help….When scaffolding is provided by the teacher, it involves the teacher in executing parts of the task that the student cannot yet manage. A requisite to such scaffolding is accurate diagnosis of the student’s current skill level or difficulty and the availability of an intermediate step at the appropriate level of difficulty in carrying out the target activity. Fading involves the gradual removal of supports until the students are on their own. (A. Collins et al., 1991, p. 14)

Step size is also managed in cognitive apprenticeship by controlling the amount of complexity in the tasks the learner is required to perform:

Increasing complexity refers to the construction of a sequence of tasks such that more and more of the skills and concepts necessary for expert performance are required (VanLehn and Brown, 1980; Burton, Brown, and Fisher, 1984; White, 1984). For example, in the tailoring apprenticeship described by Lave, apprentices first learn to construct drawers, which have straight lines, few pieces, and no special features, such as waistbands or pockets. They then learn to construct blouses, which require curved lines, patch pockets, and the integration of a complex subpiece, the collar. There are two mechanisms for helping students manage increasing complexity. The first mechanism is to sequence tasks in order to control task complexity. The second key mechanism is the use of scaffolding, which enables students to handle at the outset, with the support of the teacher or other helper, the complex set of activities needed to accomplish any interesting task. (A. Collins et al., 1991, p. 15)

Table 4 summarizes the local principles from the theories reviewed that are subsumed by the universal principle of step size.


Table 4

Principles of Learning Subsumed by the Universal Principle of Step size

Theory Group   Local principles




Graduated attainment



Gradually all non-successful impulses will be stamped out

Formation of each association may be represented with a time curve

Associative shifting

The extreme of ease: when a single experience stamps the association in completely

Improvement is the addition or subtraction of bonds

Changes in rate of improvement result from the number of bonds and the ease of formation



Each reinforced pairing of CS and UCS is a step up in strength of association

Learning steps as progressive differentiation of similar stimuli



Gradually, and usually very gradually, the number of right responses exceeds the number of wrong responses until finally all the responses are right and the habit is formed






Developing competence in any field must be divided into a very large number of very small steps



Fractional anticipatory goal reaction

Step-size as a function of generalization and stimulus condition equivalence



One trial learning as the smallest step of progression

Learning of skills requires the acquisition of many movements



Number of new elements conditioned on each sampled trial

Number of new elements must be significantly less than the number of elements   already conditioned to the response



Cognitive   Ebbinghaus:

Number of syllables that can be correctly recited after only one ready (usually seven)

Number of repetitions necessary for the memorization of a series increases with extraordinary rapidity with the increase in number of the syllables



No specific correlating local principles identified



No specific correlating local principles identified


Cognitive Information Processing:

Limited capacity of attention

Limited capacity of short-term memory

Rehearsal buffer of limited size



Some tasks are so complex that they cannot be learned directly

Conditions of practice should gradually begin to approximate the desired (unprompted) end-point of the learning product

Step size of practice trials should be adjusted to learner’s success or failure on preceding learning

Progressive differentiation


Schema Theory:

Tuning: minor modification to bring established categories into congruence with the functional demands placed on them

Restructuring: results from unwieldiness and ill-formedness of some critical mass of information that has been


Constructive   General:

Too small of a step size decontextualizes new information



Development is a continuous and smooth process



Acts organized into ensembles

Larger and larger units of information

Simple components of complex operations

Growth in spurts



Human   Self-Efficacy:

Graduated building up of self-efficacy through progressively attenuated scaffolding






Incremental attainment through repeated exposure

Zone of proximal development

Good learning is in advance of development



Progressive response induction aids

The number of subskills which must be combined, or learned to perform a more complex skill


Situated learning:

Gradual increase in responsibility


Activity theory:

Large-scale cycles involve numerous smaller cycles of learning actions


Cognitive apprenticeship:

Managing step size through scaffolding and fading

Increasing complexity



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