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Tuesday, November 08, 2005




Outline of Work to be done on

Learning Theory and the Core Liberal Arts Curriculum


Part Two


Systems of Mathematics and Ontological Models


An example of a system of mathematics is arithmetic. As mathematics majors know, a discipline called “elementary number theory” is a full system of arithmetic when seen in its entirely. A simple part of arithmetic can be used as an introduction to the foundation of mathematics. In order to better make this presentation to liberal arts students, the mathematician may refer to arithmetic as a door to higher mathematics rather than as a skill that is needed to make change in a retail store.


Personal discipline is applicable to learning abstraction, ways of reasoning, and facts found within any elementary system of mathematics.


Our strategy is a strategy based on self-discovery as opposed to memorization of facts.


In one approach, an “ontological model” is developed by the class, or individual, about the nature and construction of a system of arithmetic. The model of arithmetic is constructed using a set of ordered triples:


< a, r, b >


where a and b are topics of interest and r is a relational category. This model is the result of class participation in a pedagogy of discovery.


The topics in any system of mathematics can be usefully categorized into one of three sets:


{ topics that I know }

{ topics that I do not know }

{ topics that I do not know that I do not know }


Topics can also be seen to be easy or difficult.


In our strategy, the listing of the topics of a system of arithmetic is the instructor’s first request of the student. The student is also asked to write a short exposition about how he or she feels about mathematics.


The listing of topics serves many purposes.


The list may be made into a set


A = { a j }


and the relationship categories specified later on.


The relationship categories can be fun to find, and as they are found; there can be personal engagement in framing the entire curriculum as something “with defined boundaries”. The lack of definable boundaries in arithmetic and algebra learning may be one of the primary obstacles placed in front of the student.


The set of topics can be written one at a time onto standard index cards. Students are asked to develop the list ON THEIR OWN. After demonstrating that they have a list (and an essay), students can compare their list with others. Failure to participate is deemed to be the only wrong answer. It is critical to separate those students who resist participation from those who are trying to develop a new orientation that opens access to mathematics for them. The difficulties are often behavioral and means must be in place to allow those who want to move forward to do so [1].



It is important to note that the categorization of topics is dependant on the time in which the categorization occurs and mood of the student when the categorization occurs. For example, something that is difficult may be at a different time easy for the same student.


If the difficulty and the easiness of a topic are experienced clearly; then several types of learning may occur. This clarity comes from participation in the listing process and in personally imparting legitimacy to the process of learning.


Learning about learning is the most important consequence to participation. Learning about learning has rewards beyond the freshman mathematics class. For example, learning specific topics may be illusionary. An individual may believe that a topic was understood when in fact the topic was not understood. An individual may feel that a topic is not understood, when in fact the individual does understand the topic (but is confused by something else).


The class as a whole will develop an “ontological model” of the topics within a curriculum and this model will be seen as dynamic. The development of an ontological model of individual and class understanding of the curriculum is an example of “knowledge management” and ties into a scholarly literature on the management of knowledge within a business or social unit.


At core to the alternative curriculum is the question


What does it mean to understand ?”\




[1] The management of the classroom may require that a specific class be re-organized after the class starts. In depth behavioral/psychological testing as well as skill based testing might help reduce this need for re-organization. However, this type of in depth testing is generally not available.