What Is Mathematical Psychology?

Mathematical psychology is a branch of psychology that uses mathematical models to study psychological phenomena. Quantitative methods to describe psychological phenomena. Due to the complexity of the research objects, generally only a fuzzy qualitative description of the research can be made. However, driven by the development of information theory, cybernetics, statistical decision theory, and computer science, mathematical psychology has also developed rapidly and has continuously achieved new results. . The advantage of using mathematical models to describe psychological phenomena is not only that it is more general, accurate, deductive, and predictive than natural language descriptions. It is more important that it is easy to simulate with computers and create for the study of artificial intelligence. Better assistance.

1860 Germany

Generally speaking, the establishment of a mathematical model is first of all to separate the psychological phenomena that need to be studied, such as perception, learning, decision-making, etc. from complex psychological activities to form a specific set, and process the original data into sets. Object and relationship. Then express them in the form of algebraic, geometric, probabilistic, axioms, or in the form of computer programs and equations. Here, the main problem is to determine the correspondence between the empirical system of the research field and the formal system that expresses it.

After the mathematical model is established,

Behavioral Theory of Mathematical Psychology

In the first half of the 20th century, behaviorism was dominant, and its basic position was that learning and research should not involve psychological processes that are impossible to observe, but only limited to visible behaviors. Such research is scientific. The American psychologist Thorndike is a representative of behaviorism. He proposed a learning theory with the main characteristics of "stimulus-response connection" and "trial and error". He believed that learning is to form a stimulus-response connection. This connection is direct. The non-intermediary is formed through repeated attempts (continuously rejecting false reactions and retaining correct responses). He proposed three learning laws based on experiments: the law of preparation, the law of practice, and the law of effect. In his 1922 book, The Psychology of Arithmetic, he pointed out that arithmetic learning is nothing more than a set of special behavioral habits directed at a certain quantity and relationship. Thorndike's views provide a basis for mechanical exercises and training in mathematical learning. Skinner, another representative of behaviorism, further developed the idea of behaviorism and put forward the principle of operational conditions. He believes that simple practice cannot guarantee the recurrence of behaviors. Formation depends on strengthening. Therefore, he put forward the "stimulus-response reinforcement" learning model, and designed teaching machines and procedures for teaching. Skinner's theory has laid a certain foundation for the future development of educational technology.

Cognitive theory of mathematical psychology

In the second half of the 20th century, with the deepening of the study of learning psychology, the serious shortcomings of behaviorist ignorance of the internal psychological process of learning have become increasingly apparent. More and more psychologists have turned to focus on the internal process of learning, which has contributed to cognition. The Formation of Doctrine of Learning.

Gestalt in Germany is an early representative of cognition (Gestalt is a German word meaning Gestalt), and its core characters are Wertheimer, Kaufka, Kohler, and others. This school of thought holds that thinking is an overall meaningful perception. They take "gestalt" as the basic concept, emphasize the understanding of the nature of learning as a whole, and put forward the theory of epiphany learning. It was also Tolman who influenced the formation of cognitive theory in the early days. The idea of "intermediate variables" (that is, the "internal mechanism" of the learning subject) that he proposed became the core concept of his learning theory.

Swiss psychologist Piaget is an important representative of contemporary cognitiveism. He has conducted in-depth research on the occurrence and development of psychology, cognitive structure and its functions, and put forward the famous cognitive construction theory and cognitive development. theory. "Operation" (that is, thinking operation) is a key concept in Piaget's theory. Based on this, he divides children's cognitive development into four main stages, namely the sensory-motor stage, pre-operation stage, specific operation stage and formal operation stage. The basic characteristics and interrelationships of cognitive development in each stage are discussed. Piaget put forward the concepts of "assimilation" and "adaptation" in the book "Principles of Genesis of Epistemology", which are widely used to explain cognitive development in learning. He paid particular attention to the unique psychological characteristics of mathematical learning, and he even used mathematical methods to define some concepts in his cognitive theory (such as thinking structure, reflexive abstraction, etc.).

Mathematical Psychology Mathematical Learning Theory

Beginning in the 1960s and 1970s, cognitive theory in mathematical learning theory has replaced inevitable behaviorism. Bruner puts forward the theory of discovery learning, emphasizes that the learning process is a positive cognitive process, and advocates discovery learning of knowledge. He carried out a large number of mathematical learning experiments, and summarized four mathematical learning principles, namely, the construction principle, the symbol principle, the comparison and variant principle, and the correlation principle. In addition, Ausubel proposed the "meaningful learning" theory, and Gagne proposed the "information processing" learning theory. It is precisely because of the emergence of many cognitive learning theories that the paradigm of mathematical psychology has undergone an important change, and it indicates that there will be new developments in cognitive theory.