1. Cognitive Psychology in Mathematics Education Contributor© POSbase 2004 The overview of Anderson, Reder, & Simon (2000).Anderson, Reder, & Simon (2000). Cognitive psychology has been criticized from proponents of two other movements: (1)Situated learning (2)Constructivism Anderson et al. examined the central claims by the two movements and responded to their criticism. We first review the claims for situated learning and then for constructivism.

2. Cognitive Psychology in Mathematics Education: Situated learning © POSbase 2004 Claim 1: Action is grounded in the concrete situation in which it occurs. This claim got some support from research by Carraher et al. (1985) that demostrated that streetchildren in Brazil can multiply numbers when it is within a situation of selling lemons, but not when it is a school-type task. Therefore, action must be grounded in the concrete situation. This is the case in some situations, but not in others. E.g., carrying is bound to a context of doing base-ten additions and does not generalize to other situations. However, other tasks do generalize, such as reading and much of school mathematics, and instructors often do not need to teach knowledge in the precise situation where it is needed. Knowledge seems to be more context-bound if it taught in only one context. Therefore, some variability in context helps acquire new knowledge about a domain.

3. Cognitive Psychology in Mathematics Education: Situated learning © POSbase 2004 Claim 2: Knowledge does not transfer between tasks. This claim is a corollary of the first claim that action is situated and got support from findings that transfer between tasks did not occur. However, research shows that it depends on the experimental situation and the relation of the original task to the transfer task whether large amounts of transfer, modest amounts of transfer, no transfer, or even negative transfer is observed. Other factors influencing transfer are problem representation, degree of practice, the problem domain, and number of shared symbolic components between the original problem and the transfer task.

4. Cognitive Psychology in Mathematics Education: Situated learning © POSbase 2004 Claim 3: Training by abstraction is of little use; real learning occurs in "authentic" situations. Abstract instruction is useless if it does not fit the needs of the job to be done, leading to practitioners‘ advice to newcomers to „forget all what you have learned“. However, abstract instruction has been shown to be powerful, e.g., in learning to sex chicken, where novices can attain expert level within 20 minutes (Biederman & Shiffrar, 1987). Most effective is a combination of abstract instruction and providing concrete illustrations, as in learning to throw darts at a target underwater (Judd, 1908).

5. Cognitive Psychology in Mathematics Education: Situated learning © POSbase 2004 Claim 4: Instruction needs to be done in a highly social environment. While it is important to learn social skills in a social environment, the acquisition of other skills in a social environment would be a waste of time. It has been shown that it is at least as efficient to learn component skills individually, and careful analyses of cooperative versus individual learning have shown that there is no advantage (but often substantial costs) of cooperative learning. Part of the costs are due to motivational factors, others to coordination and to different states of ability.

6. Cognitive Psychology in Mathematics Education: Situated learning © POSbase 2004 Summary: Situated learning is strong where it considers principles of cognitive psychology, but has its weaknesses where it disregards them. Cognition is partly context-dependent, but partly context-independent. There are both successes and failures of transfer, and abstract instruction may help. Often, individual instruction is more efficient. Some proponents of situated learning claim that learning by doing in the real-life domain is the optimal procedure. From a motivational view, this is certainly not true when it comes to life-threatening tasks (firefighting), when the situation is infrequent and unpredictable (what to do as a pilot if birds hit the plane during take-off), or when the situation is potentially embarrassing (speaking a second language as a beginner).

7. Cognitive Psychology in Mathematics Education: Constructivism © POSbase 2004 Claim 1: Knowledge cannot be instructed (transmitted) by a teacher, it can only be constructed by the learner. While learning involves activity by the learner, and not only by the teacher, students need to be instructed what they can not find out themselves. Learning by discovery has been shown often to be inferior to more traditional instructional methods. When constructivists often equate cognitive psychology with drill. This is not the case: Learning-by-doing theories are widely employed by cognitive science and include analyses of how cognitive structure accommodates to experience.

8. Cognitive Psychology in Mathematics Education: Constructivism © POSbase 2004 Claim 2: Knowledge cannot be represented symbolically. This claim comes from a narrow view of symbolic representation that is expressed as words, sentences, or equivalent formal structures. In addition, it is claimed that symbolic representation is an accurate depiction of the outside world. However, symbolic representation can be pictorial, and it can be an inaccurate depiction of the outside world. Cognitive competence in a domain depends on the availability of symbolic structures that are created in response to experience. Research has shown that humans are able to manipulate fairly complex symbolic systems.

9. Cognitive Psychology in Mathematics Education: Constructivism © POSbase 2004 Claim 3: Knowledge can only be communicated in complex learning situations. There may be reasons to study skills in a complex situation, e.g., motivation or if the skill can be trained exclusively in the complex situation. However, in most situations, it is smart to train the component skills before it is trained in the complex situation. An obvious example is a symphony orchestra: If the individual players can not play the piece individually, it does not make sense to exercise as an orchestra.

10. Cognitive Psychology in Mathematics Education: Constructivism © POSbase 2004 Claim 4: It is not possible to apply standard evaluations to assess learning. Proponents of constructivism claim „that students are the best judges of what they find problematical and encourages them to construct solutions that they find acceptable given their current ways of knowing“. However, many concepts that children come to naturally are plain wrong (e.g., that earth is a disc; that the sun rotates around earth), and taking constructed knowledge of students as criteria for what should be learned may lead to inferior knowledge.

11. Cognitive Psychology in Mathematics Education: Constructivism © POSbase 2004 Summary: In contrast to the claims of constructivists, knowledge can be transmitted by a teacher, it can be represented symbolically, and does not need to be communicated in complex learning situations. Moreover, it seems wise to set some standards that are independent of the standards of the learners in order to evaluate learning. In sum, it has been shown that instructional techniques derived from constructivist claims, such as discovery learning, are often less effective and less efficient than instructional methods based on knowledge about cognitive processes.