To be a student is to achieve an understanding about the relationship of self to knowledge. In Europe students study philosophy as an integral component of their education. An understanding of philosophy enhances every step of the process of education. Philosophy drives learning as it respects and guides a student’s critical discourse with the disciplines. Students become greater purveyors of what is before them.
The structure of the knowledge that students encounter and the tools that students must employ to effectively engage with that knowledge vary. Knowledge and methods of engagement are not all apples. Students must possess both competence in the structure of knowledge and respect for the strengths and weaknesses of tools that can bring them closer to that knowledge. Mathematical and scientific understanding require one set of skills while poetic and literary understanding require another, though not mutually exclusive, set.
Foundations of knowledge acquisition for the past thirty or more years that were fowarded by Robert Gower and Jon Saphier, Howard Gardner, Robert Swartz and Jay McTighe all point to the need for respect for the nature of the disciplines with which students engage.
In this context, students wishing to gain true understanding must be ready to look behind the face of assignments and expectations This approach may not be the wish of those who say “just tell me how to do it” but it is the way by which students acquire knowledge that will never be lost.
The study of mathematics includes algebra, plane geometry, trigonometry, analytic geometry, behavior of functions, discrete analysis, multi-variate analysis, understanding sequences and series, calculus, statistics, theoretical algebra and computer programming and modeling.
Computation is not mathematics. Computation is quick counting. It provides a way to synthesize numbers or take them apart. Students should move through computation rapidly and not “explore” the trivial. Students will sometimes rely on exploratory techniques with which they have found success and then fail to move on to more efficient algorithmic techniques.
Separating the syntax of the language of algebra from the concepts and techniques of solution is critical. Syntax must be executed regularly and with precision. Drilling is a necessity to assure competence in the manipulation of expressions. It is impossible to move into higher levels of technique if the fundamentals are weak and unhinge complex processing and thinking. Respecting the need for precision is a key element in succeeding.
Thinking mathematically is visualizing relationships and utilizing symbolic language to describe and analyze those relationships. Understanding the language of algebra involves knowledge of the rules of manipulation of symbols. The techniques of algebra facilitate the modification of expressions that capture described conditions, reduce their complexity and provide a simplification and/or solution to the described condition.
The language and techniques of algebra facilitate the understanding of one, two and three dimensional phenomena such as finding x (as the saying goes), understanding linearity or curvilinearity and two dimensional and three dimensional spaces to name a few. They permit the exploration of the qualities of relationships whose characteristics can be visualized within these dimensions. Algebraic language and technique can capture the notion about how changes in one element (sales, for example) affect expectations for another (profit.)
Learning, utilizing and conveying mathematical ideas and processes have a structure similar to the features of communication. Sequencing events is necessary in good story telling and so is addressing causality. In communicating we create a picture. What does the picture say? What is behind the picture? What affects the actions and reactions of elements in the story. All of these are elements in exposing, understanding and interpreting the consequences of a mathematical condition.