Instructional versus schedule control of humans' choices in situations of diminishing returns

Four adult humans chose repeatedly between a fixed-time schedule (of points later exchangeable for money) and a progressive-time schedule that began at 0 s and increased by a fixed number of seconds with each point delivered by that schedule. Each point delivered by the fixed-time schedule reset the requirements of the progressive-time schedule to its minimum value. Subjects were provided with instructions that specified a particular sequence of choices. Under the initial conditions, the instructions accurately specified the optimal choice sequence. Thus, control by instructions and optimal control by the programmed contingencies both supported the same performance. To distinguish the effects of instructions from schedule sensitivity, the correspondence between the instructed and optimal choice patterns was gradually altered across conditions by varying the step size of the progressive-time schedule while maintaining the same instructions. Step size was manipulated, typically in 1-s units, first in an ascending and then in a descending sequence of conditions. Instructions quickly established control in all 4 subjects but, by narrowing the range of choice patterns, they reduced subsequent sensitivity to schedule changes. Instructional control was maintained across the ascending sequence of progressive-time values for each subject, but eventually diminished, giving way to more schedule-appropriate patterns. The transition from instruction-appropriate to schedule-appropriate behavior was characterized by an increase in the variability of choice patterns and local increases in point density. On the descending sequence of progressive-time values, behavior appeared to be schedule sensitive, sometimes even optimally sensitive, but it did not always change systematically with the contingencies, suggesting the involvement of other factors.     

Unit price and choice in a token-reinforcement context

Pigeons were exposed to multiple and concurrent second-order schedules of token reinforcement, with stimulus lights serving as token reinforcers. Tokens were produced and exchanged for food according to various fixed-ratio schedules, yielding equal and unequal unit prices (responses per unit food delivery). On one schedule (termed the standard schedule), the unit price was held constant across conditions. On a second schedule (the alternative schedule), the unit price was either the same or different from the standard. Under conditions with unequal unit prices, near-exclusive preference for the lower unit price was obtained. Under conditions with equal unit prices, the direction and degree of preference depended on ratio size (number of responses per exchange period). When this ratio differed, strong preferences for the smaller ratio were observed. When this ratio was equal, preferences were nearer indifference. Response rates on the multiple schedule were generally consistent with the preference data in showing sensitivity to ratio size. Results are discussed in terms of a unit-price model that includes handling and reinforcer immediacy as additional costs. On the whole, results show that preferences were determined primarily by delay to the exchange period.     

Students’ Reasoning With Reversible Multiplicative Relationships

In an 8-month teaching experiment, I investigated how 4 sixth-grade students reasoned with reversible multiplicative relationships. One type of problem involved a known quantity that was a whole number multiple of an unknown quantity, and students were asked to determine the value of the unknown quantity. To solve these problems, students needed to produce a fraction of the known quantity that could be repeated some number of times to make the known, rather than repeat the known quantity to make the unknown quantity. This aspect of the problems involved reversibility because students who do not make a fraction of the known quantity tend to repeat the known quantity (Norton, 2008 Norton, A. 2008. Josh's operational conjectures: Abductions of a splitting operation and the construction of new fractional schemes. Journal for Research in Mathematics Education, 39(4): 401–430. [Web of Science ®] , [Google Scholar]; Steffe, 2002 Steffe, L. P. 2002. A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior, 20: 267–307. [Crossref] , [Google Scholar]). All four students constructed schemes to solve such problems and more complex versions where the relationship between known and unknown quantities was a fraction. Two students could not foresee the results of their schemes in thought—they had to carry out some activity, review its results, and then carry out more activity in order to solve the problems. The other two could foresee results of their schemes prior to implementing them; their schemes were anticipatory. One of these two also constructed reciprocal relationships, an advanced form of reversibility. The study shows that constructing anticipatory schemes requires coordinating three levels of units prior to activity, a particular whole number multiplicative concept. The study also reveals that even students with this multiplicative concept will be challenged to construct reciprocal relationships. Suggestions for further inquiry on student learning in this area, as well as implications for classroom practice and teacher preparation, are considered.     

RISKY CHOICE IN PIGEONS: PREFERENCE FOR AMOUNT VARIABILITY USING A TOKEN-REINFORCEMENT SYSTEM

Pigeons were given repeated choices between variable and fixed numbers of token reinforcers (stimulus lamps arrayed above the response keys), with each earned token exchangeable for food. The number of tokens provided by the fixed‐amount option remained constant within blocks of sessions, but varied parametrically across phases, assuming values of 2, 4, 6, or 8 tokens per choice. The number of tokens provided by the variable‐amount option varied between 0 and 12 tokens per choice, arranged according to an exponential or rectangular distribution. In general, the pigeons strongly preferred the variable option when the fixed option provided equal or greater numbers of tokens than the variable amount. Preference for the variable amount decreased only when the alternatives provided widely disparate amounts favoring the fixed amount. When tokens were removed from the experimental context, preference for the variable option was reduced or eliminated, suggesting that the token presentation played a key role in maintaining risk‐prone choice patterns. Choice latencies varied inversely with preferences, suggesting that local analyses may provide useful ancillary measures of reinforcer value. Overall, the results indicate that systematic risk sensitivity can be attained with respect to reinforcer amount, and that tokens may be critical in the development of such preferences.