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The Grammar of Mathematics

Three interpretations of the classical trivium (grammar, logic/dialectic, and rhetoric) will be suggested: one, stages of development; two, course content; and three, elements of classical pedagogy. The stages interpretation will include defining the grammar of mathematics to be counting strategies, place value, and the composition and decomposition of units of higher value. An argument will be made that these concepts are prerequisites to understanding addition, subtraction, multiplication, and division – the four operations too often believed to be the grammar of mathematics. Then teachers will be challenged to re-connect a too-often excessively modularized mathematics curriculum by more clearly identifying dependencies and relationships amongst concepts as students move into the logic stage of development. To complete the stages interpretation, strategies will be given for developing in students an appreciation for persuasive and eloquent mathematical compositions and more generally, the sheer beauty of mathematics.
In the interpretation of the trivium as course content, it will be noted that the majority
of schools include grammar as a subject taught across the elementary school years, two
or more courses in formal logic, and three or more rhetoric courses often culminating in a senior thesis. Teachers will be reminded of our commitment to a well integrated K-12 curriculum and thereby challenged to understand their role in developing graduates who are excellent rhetoricians. Strategies for teaching rhetoric (as the art of eloquently using available and appropriate means to persuade an audience to a right action or belief) in
the mathematics classroom will then be emphasized. Strategies will include but not be limited to integrating collaborative problem-solving, a paradigm shift from “solutions” to “compositions”, creating daily “galleries” of student work, and establishing familiarity with a canon of mathematical proofs.
In the third interpretation of the trivium grammar, logic, and rhetoric will be described as pedagogical elements present in every lesson at every grade level. It will be shown that contemplating the presence of these elements (intentional or otherwise) while designing and implementing lesson plans will make instruction more effective.
Specific strategies will be given for reforming a K-12 mathematics curriculum. Particular attention will be given to four critical components of reform: curriculum mapping, re- connecting an excessively modularized curriculum, refining pedagogy, and re-educating the parent community.