By Crean

It is my will to establish a brief discourse in radical constructivism, and relate this to beliefs of self, specifically with regards to mathematics (and extrapolated to the general teaching arena).

Note: The constructivist perspective may be considered extremely "revolutionary" in traditional academic arenas. Although constructivism has gained a powerful ally with most modern views of mathematics, it is still rejected by a good portion of traditionalists.

• You are to close your eyes, when I tell you.
• You are to listen to what is put on the radio.
• There is some linkage between what you are listening to and the lecture.
• Throughout, try to deduce the relevance of the listening with the presentation.
• Write these down in your note-book throughout the presentation.
• The Audio Tape is a repeated sampling ("time becomes" -orbital)

"Nothing is true; everything is permitted" – Hassan I Sabbah

"…constructivism intends to undermine too large a part of the traditional view of the world. Indeed, one need not venture very far into constructivist thought to realize that it inevitably leads to the contention that man – and man alone – is responsible for his thinking, his knowledge and, therefore, also for what he does. Today, when behaviorists are still intent on pushing all responsibility into the environment, and socio-biologists are trying to place much of it into genes, a doctrine may well seem uncomfortable if it suggests that we have no one but ourselves to thank for the world in which we appear to be living." (Glasersfeld pg. 193)

Knowledge can never be interpreted as a picture or representation of the "real world", but only as a reflection of different pathways to understanding. (Glasersfeld, pg. 194)

The radical comes in when relating to knowledge. "Where as the traditional view of epistemology, as well as of cognitive psychology, that relation is always seen as a more or less picture-like (iconic) correspondence or match, radical constructivism see’s it as an adaptation in the functional sense." (Glasersfeld, pg. 196)

• Glasersfeld goes on to describe a functional relationship with Darwin’s survival of the fittest model. We are continually existing in a competition for reality. We are competing for a world view, a consensual reality.
• "Radical constructivism, thus, is radical because it breaks with convention and develops a theory of knowledge in which knowledge does not reflect an ‘objective’ ontological reality, but exclusively an ordering and organization of a world constituted by our experience. The radical constructivist has relinquished "metaphysical realism" once and for all, and finds himself in full agreement with Piaget, who says: ‘Intelligence organizes the world by organizing itself." (Glasersfeld, pg. 199)

Please think on the full implications of this. In summation, reality is relative, there is no way to know an absolute reality (even if it did exist) so recognize that, only the things to enter into the realm of the experience are to be considered real (if it affects us, it is real). We construct our knowledge as it fits into our world-view.

A model of instruction is proposed with six components (Confrey, pg. 107):

• The promotion of student autonomy.
• The development of reflective processes.
• The construction of case histories.
• The identification and negotiation of tentative solution paths (work with students).
• The retracing and group discussion of the paths used to solutions.
• Adherence to the intent of the materials (We are not to go off on tangents).

"There has recently appeared in increasing amounts of evidence that direct instruction may not provide an adequate base for students’ development and for student use of higher cognitive skills." (Confrey, pg. 107)

Definition: Direct Instruction – Constituent of three parts according to Confrey.

1. Relatively Short Products are expected from students, rather than process oriented answers. (Note how prevalent this is in a good portion of academics)
2. Teachers, for the most part, simply execute their planned routines. Teachers rarely operating outside of given bounds.
3. The responsibility for determining what level of learning the student has reached lies solely with the teacher.

In relation to mathematics: mathematics is not a tool to learn how to use, but a thought process to develop ones cognition (Confrey, 1990).

A model of a social cognitive learning theory

Personal factors include beliefs and attitudes that affect learning, especially in response to environmental stimuli.

Behavioral factors include responses one makes in a given situation.

Environmental factors include the role of people which are in the environment of the person. (Textbook, ch. 6 pg. 129).

The chapter goes on to elaborate on different aspects of this theory. It’s main emphasis for teaching is to try and exemplify positive attributions and retrain undesirable attributional patterns. The main emphasis of the chapter is on Self Efficacy.

• Encourage students constructively.
• Monitor feedback.
• Set specific goals.
• Expect that the students will succeed.
• In general be positive, have realistic expectations and, believe in your students.

Note: Encourage individual autonomy also. One study has show that "many middle and secondary teachers communicate an authoritarian and thereby limited view of mathematics" (Cooney, 1998). I am sure this is not only in regards to mathematics. One article by Cooney, Shealy, and Arvold dictates that it is highly beneficial to adapt a continual analysis of our personal belief structures as teachers to that we do not get too static in our views. This is in accordance with a constructivist perspective.

Cognitive psychologists and mathematicians have been found to agree on certain underlying principles of knowledge restructuring (Seldon, 1997). Note that these principles can be abstracted to almost any field of knowledge that is interactive.

• Knowledge is acquired by construction, not transmission alone.
• Knowledge acquisition involves restructuring (as new material is learned).
• The process of knowledge acquisition is restrained by what one already "knows".
• Knowledge is domain specific (also interactive with self efficacy).
• Knowledge acquisition is "situated" (see encoding processes chapter 4).

Try and observe these when lesson planning, exam planning etc.

Get together with the person next to you and discuss what each of you thought the connections were between the presentation and the beginning experiment.

Confrey, J. (1990). What constructivism implies for teaching. In R. B. Davis, C. A. Maher & N. Noddings (eds.), Constructivist views on the teaching and learning of mathematics (JRME monograph 4) (pp. 107-122). Reston, VA :NCTM.

Cooney, T; Shealy, E; Briget Arvold. Conceptualizing Belief Structures of Preservice Secondary Mathematics Teachers. Journal for Research in Mathematics Education (1998, Vol 29, No. 3. 306-333).

Glasersfeld, E. An Introduction to Radical Constructivism. (Chapter 10), from the book The Construction of Knowledge (Glasersfeld, E.). Mention of Piaget (1937) La Construction Du Reel Chez l’Enfant. (p. 311).

Green, Thomas F. (1971).The Activities of Teaching. NewYork: McGraw-Hill Book Company. (Chapter 3) Teaching and the Formation of Beliefs.

Selden, John & Annie (1997). Should Mathematicians and Mathematics Educators be listening to Cognitive Psychologists? MAA online (The Mathematical Association of America) Research Sampler. http://www.maa.org/t_and_l/sampler/rs_2.html