This fast-paced learning experience leaves little time for exploration, experimentation, discovery, and argumentation – the processes that were necessary for the development of mathematics across history and are still at the very core of what real mathematicians do. High school geometry is the one course with the greatest potential to engage students in these processes through the challenge of writing mathematical proofs. Yet geometry students are rarely given the time to observe, induce, form, critique, refine, organize, and informally express relationships between numbers and figures. Instead, students are immediately held to a rigid two-column format requiring the very deductive reasoning skills that were neglected and avoided in prior courses. For the students’ sake teachers often limit their consideration to a body of proofs requiring a very limited repertoire of predictable and easily memorized deductive maneuvers.
Thinking creatively and critically, assessing what you don’t know, what you do know, remembering methods/approaches that have worked in the past, anticipating the skills you might use, forming an initial strategy of attack, stepping forward, assessing progress, stepping back then forward again, etc. – these are the experiences that make a study of mathematics most valuable to other disciplines. When these experiences are neglected teachers are then limited to justifying a study of mathematics through immediate real-world applications represented in the form of “word” or “story” problems. Yet the applications are over-simplified, the contexts are far removed from the student’s experience, and the “story” problems rarely tell compelling stories. For many students, word problems such as these simply become annoying syntactic translation exercises that are hardly motivating.
In order to repair mathematics instruction in the United States, teachers must first recognize the primacy of their own content knowledge. If a teacher’s mathematical understanding is fragmented, excessively procedural, narrow, shallow, and decontextualized, then that teacher cannot reasonably expect the understanding of his/her students to become any different. In order for a teacher to navigate his/her way through oversized textbooks, identify and emphasize essential content, design lessons around meaningful learning objectives, engage students in critical thinking, appropriately represent and value the struggles of problem solving, and teach mathematical reasoning he/ she must know the content well – not just the content of a single grade level or course but rather the entire K-12 curriculum.
Should the teaching practices identified by recent research be corrected by teachers recommitted to increasing their content knowledge and refining their pedagogy, the United States might rise in the rankings but the mathematical experience of students would still remain incomplete. Why? Because our obsession with international competitiveness has caused us to completely neglect those immeasurable characteristics of mathematics that make mathematics most lovable. Coherence and contextualization must be gained through the integration of the historical narrative – its master works and colorful personalities. Students must have their attention drawn to universals encountered in mathematics, truths indicative of something eternal and greater than the mathematics itself. Time must be devoted to contemplation of the sheer beauty of mathematics. Mathematics must be recognized as a language inexplicably able to describe natural phenomena, enabling us to better understand God’s creation and praise Him for His majesty.
Let us repair these ruins.