Mathematics and Logic - Skill and Concept Development

Appendix A. Story Telling
Key Talent

The ability to tell, invent and follow stories is transformed in education and research into the ability to tell, invent and follow explanations and instructions as a means to enrich, advance and provide a context for observable skill development, and a means to share what is the mind. In essence telling stories, one step at a time, and one step after another (in order perhaps) is the key to knowledge of folklore, culture and technical knowledge. We may judge stories as fictional or not, to greater or lessor extent. We judge the consistency of stories in accordance with our knowledge or view of what is possible or allowable. So a good alibi allows a detective to ease or remove suspicions about who committed a action. In writing or inventing a story, an author will attempt to avoid immediate or implied inconsistencies. What Louis Carroll's Alice sees in Wonderland or Through the Looking Glass provides an story to follow, one that is not realistic. Shakespeare and Moliere through their writing create pieces of fiction for us to follow on paper or even to act out. We may judge the characters and plots in stories and plays by our own ideas on what is not possible. But in telling stories, the tellers provide us with a imaginary world to follow and judge. Likewise in telling theories or providing instructions for the operation of a machine or the creation of a product, researchers and teachers are telling us stories, ones that may correspond to and be useful in practice. The composers of a story may adhere to rules and criteria for consistency, rules and criteria that say what is or should be possible or not, in order to decide or conclude what may be in the story or not. Chains of reason may be employed to find what a story implies or requires. The composer may include or not, those implications or requirements in the telling and extension of a story or theory.

The developers of empirical theories in science and technology tell many small technical stories which individually work and are practice in special cases, but not necessarily in all cases. High school and college chemistry for example include alternative theories for the identification of acids, salts and bases. The oxymoron acid-salt reflects the idea that what is an salt in one theory is an acid according to another (a litmus paper test say).

reality, imagination and fiction - exploring ideas, continued

The author of a story in a book or a play creates an imaginary world for us to visit in our minds. The story may or not be consistent with our knowledge of real life. More and less can be suggested in a story than occurs in real-life. Stories can be fictional, half-fictional, approximately true to life or true.

Stories may explain or describe how things came to be. Stories may provide lessons a for reader directly or through the words and interpretations of another. Stories may give us ideas of what to do or not. Stories have plots and chains of events or reasons to follow, real or not. Stories presented on stage as plays may include not only words but also actors and props to make the plot or reenactment easier to follow. Actors have scripts to follow. Actors are defined by their names, costumes and actions.

Most of us, many of us, have the ability to follow a story, its sequence of scenes with words and events, and to recognize what is real or pretend. We can learn stories, invent them and tell them to others via spoken and written words. Stories can be told and retold in ways that are almost repeatable and reproducible. Our knowledge of a culture may come from its stories and myths.

In cooking and construction, plans and recipes give or suggest sequences of steps or actions to take to arrive at results. The steps and the results should be repeatable and reproducible. Technical know-how is based on rules and patterns to follow plus some judgment as to when they can be applied. Trying to apply rules and patterns when items they require are missing usually leads to bad results.

In mathematics, science and technology, as in daily life, there are stories to follow. These stories, normally called theories, describe a situation (say what is what is assumed) and describe as a well assume methods for arriving at results or conclusions in a step by step way. The authors of these stories or theories would like their consistency with reality. A theory is inconsistent with reality if it says two exclusive events occur at the same time or if predictions based on it fail. Unfortunately, the author of theory to say what should happen may capture a pattern in theory that works in some circumstances, but not all. So a theory may be applicable and sufficiently consistent with reality to be useful in some circumstances - those it reflects - while failing in others.

Knowledge in mathematics and science and technology is based on theory and practice. A method or procedure describe in a lab or controlled circumstances how following certain steps will give a result. Those steps and the results, done carefully enough, appear to give repeatable and reproducible independent of the doer. Methods that work in practice may be described and accumulated, and used one at a time and one after another to follow steps and arrive at results, one at a time and one after another. A skilled practitioner may recognize when one method can replace another because it gives the same result or a more convenient result.

Geometry was codified in the works of Euclid, about 300 B. C. The codification consisted of assumptions or definitions about points, straight lines, circles, triangles and the geometric figures composed from the latter. The resulting theory or theories was presented not on stage, but on paper (a prop) with the aid of rulers and compass (more props) to provide construction methods and to suggest and describe results and conclusions one at a time and one after another. Students and teachers and philosophers could follow explanations one at a time and one after another in way that follow some or all of the strands of thought in Euclid's work. The codification provides a mechanical knowledge of geometry because each of us in following the steps should verify that the steps are valid, that the implication rules used in each step are justly applied.

The foregoing gives rule- and pattern-based chain of reasons independent of the followers and authors. All that provides a model for making and arriving at conclusions with rules and patterns not only in geometry, but also in other disciplines where rules and patterns are valued as guides. But this model for reason has its limitations.

Rules and patterns describing what we have observed, drawn from experience, are not absolute. We do not know if they are fully reliable, or we may not precisely when they apply, if at all. When rules and patterns are not reliable, a risk appears. What they suggest, one at a time and one after another, may not be consistent with reality. None the less, recognizing rules and patterns in a subject provides a means for accumulating know-how for arriving at results, and a limited know-why. The latter is given by the chain of reason or suggestion with rules and patterns, reliable or not, that led to a result. (Implication rules and suggestions in a theory may themselves rely on the need for a theory to be consistent. See above).

Selby A, Volume 1A, Pattern Based Reason, 1996.

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