Formal or Informal Peer Review

Or, pne- and Two-Way Conversations with Society in the Individual Construction of Knowledge - the scientific method as a form of social (joint) constructivism

Educational theorists  may enjoy the follow perspective on the individual and social construction of mathematical skills and knowledge

For each of us mathematics is or should be a static and/or growing collection of rules and patterns involving notation, geometry and logic which can be used and combined in a repeatable, reproducible, recorded (or described) and thus verifiable manner to arrive at numerical results or further rules and patterns through calculation and/or some rules and patterns logic. This collection may grow in a rigorous manner through the addition of numbers, rules or patterns, explicitly assumed, for better or worse, and through the introduction of  further numbers, rules and patterns that are tested in the following sense. The new numbers,  rules and patterns have to be implied by calculations or reasoning which uses numbers, rules and patterns previously recognized as members of the collection, all in a repeatable, reproducible, recorded (or described) and therefore verifiable manner. In this growing individual collection of assumed and derived numbers, rules and patterns, each of may recognized certain sub- collections are more reliable than others, and certain sub-collections are more agreeable with the present and past works of colleagues through one-way or two-way social conversations with them. Here authors, living or past, communicate with each of us, through their written work. And over time, the social construction of mathematics has become a social discourse with new adherence and new directions.

As students, not quiet ready to invent or re-invent rules and patterns of arithmetic and algebra, we may be given rules and patterns to assume along with drill and practice, so that their use leads to repeatable, reproducible, recorded or well-described and hence verifiable results. Social conversation with teachers physically present or manifested through their spoken or written work may lead to the growth of a personal collection of mathematical data, rules and patterns. Again, in this growing individual collection of assumed and derived numbers, rules and patterns, each of may recognized certain sub- collections are more reliable than others, and certain sub-collections are more agreeable with the present and past works of colleagues through one-way or two-way social conversations with them.

 There-in lies a common knowledge agreeable to others and hence socially more authoritative, in which individual have become like-minded due to the manner which they accept and grow their collection or sub-collections of rules and patterns, in a repeatable, reproducible, recorded and therefore verifiable manner.

There-in lies a standard which individual need to accept for their hopes, dreams and speculation to be tested and accepted by others as part of the common knowledge.

Thus each individual has a conscience or socially acquired rules and patterns to guide and accept in  the formation of his or her personal collection and construction of knowledge. Individual departures from those social rules and patterns leads to individual perspectives of a subjective nature beyond the reach and sanction of social discourse and beyond testing. Such subjective viewpoints may be challenged by standards set in written work of others or be challenged in social discourse with others in the neighborhoods, teachers, tutors and parents included.

Over time, the social discourse in mathematics has led to a courses that present rules and patterns for students to meet and master in a repeatable, reproducible and thus verifiable manner. Answers that are not verifiable,] allow for the correction or challenge of student habits, and the possibility of more prudent or careful answers in the future. There-in lies a social discourse for the guidance and construction of a student's growing collection of mathematical rules and patterns. 

Student engagement so that they follow the guidance requires a context and motivation that may very from culture to culture  Where some cultures produce students that are potentially active or too active participants in their own education, other cultures, subcultures and times produce students who are quieter, more passive and for whom classroom procedures, even those of a constructivist nature, does not work.  The parent who does poorly in mathematics may inform his son or daughter that mathematics after arithmetic, even before, is a waste of effort. So the difficulties of one generation in mathematics, the awkwardness or inappropriateness of instruction, may be seen or ducked by the next. With students opposed to mathematics, a leaner curriculum that covers and develops key skills and concepts, those needed in practice or needed for father learning, with material that is nice to know but not necessary or not mentioned later omitted,  may provide a shorter, less alienating program. Not all is certain.

Extreme constructivism may hold that the conclusions arrived at by an individual should be respected and not challenged by an instructor. The instructor should not be an authority. Less extreme constructivism may hold that the conclusions arrived at by at a group of students should be respected and not challenged by an instructor. Again, the instructor should not be an authority. However, students in school and out learn from their environment. The environment is authoritative.  Child learn to avoid extremes of heat and cold. For better or worse, the young and aging individuals have non-verbal and then verbal interactions with their environment, and in doing so may adopt habits and customs for personal safety and survival. Nature takes care or provides  the growth - the increase in physical and mental capabilities. The development of language skills adds an iterative verbal or word-based communication to the abilities and knowledge of a child, and the customs or rules the child may learn and follow.

The child's level of consciousness may vary between visual and verbal. Each society in telling stories or providing histories provides the child or teen or adult with a greater verbal awareness and image of the surrounding environment, rules and customs included.  With this growing verbal knowledge of rules and customs, the knowledge may become less hands-on  The question of reliability appears for knowledge that is more verbal than hands-on.  There people, even a single individual, may operate or function at different levels. See the three signs of intelligence above.

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