Mindset changes lead to drastic impairments in rule finding

Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research

Individuals use diverse strategies to solve mathematical problems, which can reflect their knowledge of arithmetic principles and predict mathematical expertise. For example, '6 + 38 - 35' can be solved via '38 - 35 = 3' and then '3 + 6 = 9', which is a shortcut-strategy derived from the associativity principle. The shortcut may be critical for understanding algebra, however approximately 50% of adults fail to use it. We review the research to consider why the associativity principle is challenging and highlight an important distinction between shortcut identification and execution. We also discuss how domain-specific skills and domain-general skills might play an important role in shortcut identification and execution, and provide an agenda for future research.

Capuchin and rhesus monkeys but not humans show cognitive flexibility in an optional-switch task

Learned rules help us accurately solve many problems, but by blindly following a strategy, we sometimes fail to find more efficient alternatives. Previous research found that humans are more susceptible to this "cognitive set" bias than other primates in a nonverbal computer task. We modified the task to test one hypothesis for this difference, that working memory influences the advantage of taking a shortcut. During training, 60 humans, 7 rhesus macaques, and 22 capuchin monkeys learned to select three icons in sequence. They then completed 96 baseline trials, in which only this learned rule could be used, and 96 probe trials, in which they could also immediately select the final icon. Rhesus and capuchin monkeys took this shortcut significantly more often than humans. Humans used the shortcut more in this new, easier task than in previous work, but started using it significantly later than the monkeys. Some participants of each species also used an intermediate strategy; they began the learned rule but switched to the shortcut after selecting the first item in the sequence. We suggest that these species differences arise from differences in rule encoding and in the relative efficiency of exploiting a familiar strategy versus exploring alternatives.

Transferring control demands across incidental learning tasks - stronger sequence usage in serial reaction task after shortcut option in letter string checking

After incidentally learning about a hidden regularity, participants can either continue to solve the task as instructed or, alternatively, apply a shortcut. Past research suggests that the amount of conflict implied by adopting a shortcut seems to bias the decision for vs. against continuing instruction-coherent task processing. We explored whether this decision might transfer from one incidental learning task to the next. Theories that conceptualize strategy change in incidental learning as a learning-plus-decision phenomenon suggest that high demands to adhere to instruction-coherent task processing in Task 1 will impede shortcut usage in Task 2, whereas low control demands will foster it. We sequentially applied two established incidental learning tasks differing in stimuli, responses and hidden regularity (the alphabet verification task followed by the serial reaction task, SRT). While some participants experienced a complete redundancy in the task material of the alphabet verification task (low demands to adhere to instructions), for others the redundancy was only partial. Thus, shortcut application would have led to errors (high demands to follow instructions). The low control demand condition showed the strongest usage of the fixed and repeating sequence of responses in the SRT. The transfer results are in line with the learning-plus-decision view of strategy change in incidental learning, rather than with resource theories of self-control.

Spontaneously spotting and applying shortcuts in arithmetic-a primary school perspective on expertise

One crucial feature of expertise is the ability to spontaneously recognize where and when knowledge can be applied to simplify task processing. Mental arithmetic is one domain in which people should start to develop such expert knowledge in primary school by integrating conceptual knowledge about mathematical principles and procedural knowledge about shortcuts. If successful, knowledge integration should lead to transfer between procedurally different shortcuts that are based on the same mathematical principle and therefore likely are both associated to the respective conceptual knowledge. Taking commutativity principle as a model case, we tested this conjecture in two experiments with primary school children. In Experiment 1, we obtained eye tracking data suggesting that students indeed engaged in search processes when confronted with mental arithmetic problems to which a formerly feasible shortcut no longer applied. In Experiment 2, children who were first provided material allowing for one commutativity-based shortcut later profited from material allowing for a different shortcut based on the same principle. This was not the case for a control group, who had first worked on material that allowed for a shortcut not based on commutativity. The results suggest that spontaneous shortcut usage triggers knowledge about different shortcuts based on the same principle. This is in line with the notion of adaptive expertise linking conceptual and procedural knowledge.