This article is published with the permission of our friends at the Society for Classical Learning.
Since the launch of Sputnik in 1957 educational reform in the United States has been generally motivated by a desire to be internationally competitive. Unfortunately, cycles of reform largely characterized by document-based attempts to teacher-proof curricula through standardization and high-stakes assessment have proven largely ineffective. Recent international assessments measuring the mathematical aptitude of students continue to rank the United States below the majority of developed countries. Corresponding qualitative and quantitative studies have identified teaching practices contributing to this ongoing mediocrity. Let us now give attention to these practices so that we do not continue to repeat the same mistakes. Then let us consider those elements of mathematics that have been entirely missed due to our recent obsession with international competitiveness.
Teachers taking their cues from oversized textbooks are finding their pedagogy inevitably compromised. In an effort to increase marketability by satisfying not one but rather a multitude of published standards, the vast majority of textbooks contain far too many sections for any one teacher to “cover” over the course of an academic year. Making it to the last page of a textbook without skipping sections requires teachers to move at a breathless pace, too often achieved by limiting instruction to a series of definitions, formulas, and prescribed algorithms. Under these conditions, students engage in memorization and replication and the skills they learn are later quickly forgotten because they were never accompanied by genuine understanding.
Teachers not attempting to flip through every page of their textbook still compromise the learning experience. Their decisions to include, exclude, emphasize, or deemphasize particular content are too often based on their own narrow, shallow, and fragmented understanding of mathematics. Having often mastered only the content of a few grade levels or courses, these teachers are also unable to understand the long-term effects of their decisions. The scope and sequence that students experience reflects the strengths, weaknesses, interests, disinterests, and comfort levels of the teachers more than it does a commitment to a well-defined curriculum map or set of meaningful learning objectives. When critical or prerequisite concepts are underdeveloped and peripheral topics are over-emphasized it becomes very difficult for students to make smooth transitions from grade to grade or course to course.
Even those teachers who ground their scope, sequence, and pacing decisions upon a commitment to a core set of state and/or national content standards engage students with an anemic pedagogy. Most content standards describe skills that can easily be assessed via the multiple-choice questions of high-stakes standardized testing. Determined to check the teaching of every skill off the list, teachers tend to aim almost exclusively at meeting objectives of the student-will-be-able-to-quickly-and-accurately kind. Mathematics is then reduced to individual students choosing and swiftly executing algorithms, thereby minimizing if not eliminating experiences of investigation, adaptation, discovery, contemplation, collaboration, perseverance and creative problem solving.
American students devote anywhere from 15 seconds to 15 minutes to attempting a single problem. The AB Calculus Advanced Placement exam includes a section composed of three free-response problems to be solved in a maximum time of 45 minutes. On the other hand, for a student to attempt every problem of one mathematics section of the SAT he/she must maintain a pace of approximately 60 to 90 seconds per multiple-choice problem. The 15-minute per problem pace is an experience limited to a select group of advanced students. Most students will resist spending more than a couple of minutes on a single problem. Homework problems that cannot be solved quickly are typically set aside and asked about the next day with phrases such as “I didn’t get number 13.” The teacher typically responds with a full demonstration (and sometimes an explanation) of the proper steps then asks the student to mimic this example the next time a similar problem is encountered. The student never actually engages in any kind of significant struggle – the kind of struggle that ultimately deepens understanding and sharpens critical thinking skills by requiring students to select, adapt and experiment with various problem solving techniques.
Many students are now under challenged because they are too quickly excused from wrestling with problems. The teacher hastily concludes that the student is unable to solve a problem when the student shows no signs of knowing the first step to be taken, instead of insisting that the student make some first move, any move, even the wrong move then another and another and another until he/she has exhausted all options. This tendency to excuse students from the struggle, to save them with a quick explanation, demonstration, and/or prescribed algorithm is not present in the classrooms of top-performing nations. Instead we see students being called to the front of the classroom to attempt and re-attempt challenging problems, to receive the constructive criticism and praise of their classmates, and to persevere until a solution is found.
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.
