MAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 45 - Jerome Bruner Focus Stages of Learning Manipulatives • Pattern Blocks and Proportional Thinking • Introduction to Algebra Blocks • Algebra Tasks Other Manipulatives • Cards • Dice • Dominoes • Letter Chart • Maps • Nursery Rhymes • Sugar Cubes Discussion Jerome Bruner (1915 – present) Bruner is one of the best-known and influential psychologists of the twentieth century. He was one of the key figures in the so-called 'cognitive revolution' - but it is the field of education that his influence has been especially felt. He is one of the key players in constructivism. A major theme in Bruner’s work is that learning is an active process in which learners construct new ideas or concepts based upon their current/past knowledge. The learner selects and transforms information, constructs hypotheses, and makes decisions, relying on a cognitive structure to do so. Cognitive structure (i.e., schema, mental models) provides meaning and organization to experiences and allows the individual to "go beyond the information given". As far as instruction is concerned, the instructor should try and encourage students to discover principles by themselves. The instructor and student should engage in an active dialog (i.e., Socratic learning). The task of the instructor is to translate information to be learned into a format appropriate to the learner's current state of understanding. Curriculum should be organized in a spiral manner so that the student continually builds upon what they have already learned. Bruner (1966) states that a theory of instruction should address four major aspects: (1) predisposition towards learning, (2) the ways in which a body of knowledge can be structured so that it can be most readily grasped by the learner, (3) the most effective sequences in which to present material, and (4) the nature and pacing of rewards and punishments. Good methods for structuring knowledge should result in simplifying, generating new propositions, and increasing the manipulation of information.MAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 46 - The ideas originally outlined by Bruner originated from a conference focused on science and math learning. Bruner first illustrated his theory in the context of mathematics and social science programs for young children. “The concept of prime numbers appears to be more readily grasped when the child, through construction, discovers that certain handfuls of beans cannot be laid out in completed rows and columns. Such quantities have either to be laid out in a single file or in an incomplete row-column design in which there is always one extra or one too few to fill the pattern. These patterns, the child learns, happen to be called prime. It is easy for the child to go from this step to the recognition that a multiple table, so called, is a record sheet of quantities in completed multiple rows and columns. Here is factoring, multiplication and primes in a construction that can be visualized.” BRUNER’S EDUCATIONAL THEORY Bruner’s educational theories consist of eight. 1. Theory of Value: What knowledge and skills are worthwhile learning? What are the goals of education? 2. Theory of Knowledge: What is knowledge? How is it different from belief? What is a mistake? What is a lie? What is knowledge? Knowledge is not simply thinking and the result of intellectual activity and experience, it is the "internalizing of tools that are used within the child's culture" (GB, p. 109). It is characterized by the development of language to convey, in words or symbols, what is felt and known. Language is key to knowledge, it is the primary way that concepts can be taught and questioned. It is also the increasing ability to deal with a variety of activities simultaneously and sequentially (GB, p. I 10). How is knowledge different from belief?. I believe that Bruner would not see much difference between knowledge and belief. He felt that students learn best when their instructor leads them to discovering information on their own. As this is done, knowledge would first occur as a belief, that would later be validated by the instructor. What is a mistake? Because Bruner believes discovery learning, there is, inherently, a trial and error process that the child must go through. Mistakes are therefore simply alternative mental processes and a necessary part of learning. What is a lie? A lie, to Bruner, is anything that takes away from discovery learning, that does not capitalize on young learners who have the ability to learn anything and that does not utilize the technology and tools of our society. 3. Theory of Human Nature: What is a human being? How does it differ from other species? What are the limits of human potential? 4. Theory of Learning: What is learning? How are skills and knowledge acquired? What is learning? Learning is an active, social process in which students construct new ideas or concepts based on their current knowledge. The student selects the information, forms hypothesis and then integrates this new materialMAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 47 - into his/her own existing knowledge and mental constructs. This is a continual process. Learning occurs in three stages: 1) Enactive- in which children need to experience the concrete (manipulating objects in their hands, touching a real dog) in order to understand. 2) Iconic-students are able to represent materials graphically or mentally (they can do basic addition problems in their heads. 3) Symbolic- students are able to use logic, higher order thinking skills and symbol systems (formulas, such as 212212)()(),( yyxxBAd −+−= and understand statements like "too many cooks spoil the broth"). How are skills and knowledge acquired? These things are not acquired gradually, but more in a staircase pattern which consists of spurts and rests. Spurts are caused by certain concepts "clicking", being understood. These "clicks" have to be mastered before others are acquired, before there is movement to the next step. These steps are not linked to age but more toward environment. Environments can slow down the sequence or speed it up. Knowledge was best acquired when students were allowed to discover concepts and information on their own. 5. Theory of Transmission: Who is to teach? By what methods will they teach? What will the curriculum