A TEACHER'S PERSPECTIVE ON WHAT'S WRONG WITH OUR SCHOOLS.

• Author: McAllister, Peter

The American public has been aware of the poor performance of our schools since the Reagan Administration published A Nation at Risk: The Imperative for Education Reform in 1983. Since then, educational reformers have engaged in several major programs, such as the No Child Left Behind Act and the Common Core State Standards Initiative. Despite these efforts, the performance of our schools is still unsatisfactory. We have one of the highest levels of educational spending per student in the world, yet American students' performance in international comparisons of educational quality is consistently mediocre. According to the National Assessment of Educational Progress (2017), only 25 percent of American 12th grade students are proficient in math, 22 percent are proficient in science, and 12 percent are proficient in U. S. history.

Members of several professional groups, including economists, psychologists, teacher union representatives, public policy analysts, and politicians, have opined upon our educational policies for decades. Teachers, on the other hand, have been virtually silent. Yet teachers have direct experience of the practical consequences of our educational policies, and are in a position to identify problems in our schools that may not have been discussed adequately in the public forum.

Based upon my own teaching experience, I believe that the discouraging test performance of American students is a symptom of fundamental problems in our schools. When I taught mathematics and social studies in a public high school, I observed the following characteristics in many of my students:

* A nonconceptual mentality. Students could not integrate the information that they learned into a comprehensive understanding of a subject. Because they could not comprehend their subject matter, they simply memorized it.

* A lack of independent thought. When drawing conclusions, students rarely relied on their own informed judgment. Instead, they either relied 011 their emotional reactions or uncritically adopted the opinion of the group.

* Ethical expediency. Students frequently cut corners for the sake of some short-term gain. If they thought that the rewards were high enough, students violated rules that they would not have violated otherwise.

In this article, I argue that the policies and practices that engendered these characteristics in my students are based upon the pedagogical principles of "progressive education"--that is, the educational philosophy that has dominated mainstream American education for the past century. Although progressive education ended as a formal movement in the 1950s (Cremin 1961: 347-53), state departments of education have ensured that progressive educational principles have remained firmly entrenched within our schools. These principles, applied faithfully by American educators for generations, have produced students who do not think, who do not question, and who do not care about what is right or wrong.

Nonconceptual Mentality

Students have difficulty learning to think conceptually because many of our curricula violate a fundamental principle of epistemology: the hierarchical structure of knowledge.

In her Introduction to Objectivist Epistemology, Ayn Rand (1966) identified a crucial aspect of conceptual knowledge: concepts have a hierarchical structure. Higher-level concepts are derived from simpler, more basic concepts. For example, it is impossible for a child to grasp the concept of "furniture" without first grasping more basic concepts from which the concept furniture is derived such as "bed," "table," "chair," and so on. There is a necessary order in the sequence of abstractions that a child must follow to form this concept. This necessity of order applies to every concept. A higher-level concept, such as "electron," requires a lengthy, complex sequence of abstractions. Each step in this sequence must be performed in a hierarchical order. If any step is omitted or not performed properly, the resulting idea cannot be considered valid conceptual knowledge.

This aspect of Ayn Rand's theory of concepts has a clear pedagogical implication: because there is a necessary order in which concepts are formed, there is a necessary order in which concepts must be taught. As Lisa VanDamme (2006: 59), the headmistress of the VanDamme Academy, has pointed out, "An abstract idea--whether a concept, generalization, principle or theory--should never be taught to a child unless he has already grasped those ideas that necessarily precede it in the hierarchy, all the way down to the perceptual level." Because the order of presentation of conceptual knowledge must follow the principle of hierarchy, a proper education requires a structured curriculum that also follows this principle. If the proper order of presentation of a subject is violated, it is often impossible for a child to integrate new information into his knowledge base.

Progressive educators, however, have rejected structured, hierarchical curricula. William Heard Kilpatrick, who was reputed to have trained 35,000 teachers during his tenure at Teachers College, Columbia University (Bavitch 1983: 50), was outspoken in his antipathy toward any structured curricula. Kilpatrick (1925: 266) advocated an education that "would stress thinking and methods of attack and principles of action" rather than curricula that he derisively referred to as "subject matter fixed-in-advance." This rejection of fixed curricula led to the separation of thinking methods from content, with content relegated to a secondary status. Under the influence of Kilpatrick and other progressive educators, teachers gradually abandoned courses with content presented in a logical hierarchy.

The U.S. experience with mathematics education is an obvious example of this development. State departments of education began to mandate instruction in "New Math" in the late 1950s and, by the late 1960s, the New Math curriculum had become prevalent in our schools. As Morris Klein (1973) has pointed out, this curriculum repeatedly violated the principle of hierarchy. Elementary school students struggling to grasp the rudiments of arithmetic have had to learn aspects of number theory that are meaningless to children of their age. For example, in many schools, instead of teaching subtraction by simply counting and removing quantities of real objects, such as tiles or beads, elementary school teachers taught their students that the operation of subtraction entails the addition of a negative quantity. It took the world's leading mathematical thinkers a millennium to develop the concept of a negative quantity, and another millennium to accept it (Klein 1973: 40); yet mathematics teachers have expected children somehow to master this counterintuitive concept in a matter of weeks. Similarly, even though it took 22 centuries for mathematicians to notice that Euclid's Elements rested on more fundamental axioms than Euclid stated, geometry teachers have expected teenagers not only to grasp this shortcoming, but also to understand why they must use abstruse axioms to prove the simplest of theorems. New Math curricula have flitted across exotic, unrelated topics such as set theory, locus, symbolic logic, and linear algebra without providing students with any indication of the practical application of these topics or a logical context within which to integrate them.

To get an idea of how esoteric New Math curricula have been, consider the topic of material implication. This topic deals with conditional sentences, which are statements in the form of "if ..., then." Material implication obviates any causal relationship in these statements. Material implication holds that the only time a conditional sentence is false is when the antecedent (the "if" part of the sentence) is true and the consequent (the "then" part) is false. All other conditional sentences are true. Thus, a sentence such as "If Paris is the capital of France, then the sun rises in the West" is false, not because it makes no sense, but because the antecedent is tine (Paris is the capital of France) and the consequent is false (the sun does not rise in the West). On the other hand, the sentence "If the sun rises in the West, then Paris is the capital of France" is a true statement. Try explaining that to a class of ninth graders! Material implication first gained notoriety when Bertrand Russell included it in Principia Mathematica (Whitehead and Russell 1910: 98). Since then, philosophers have challenged the utility' of this logical formalism (Blanshard 1939: 374-81). As you may imagine, most well-educated adults would have a great deal of difficulty understanding this topic. You also may be wondering what any of this has to do with mathematics. Yet material implication was part of New York State's Integrated Mathematics curriculum for decades and appeared in widely used mathematics textbooks (Dressier and Keenan 1989: 147-51). The New York State Regents examinations regularly included questions involving material implication, or topics derived from it, until January 2009. It has also been part of the curricula of school districts in California and several other states. Material implication is not part of the Common Core standards, however, and it is no longer included in the mathematics curricula of either New York or California.

By discarding the hierarchical, logical chain that connects mathematical abstractions to physical reality, American educators have severed mathematics from the real world in the minds of many of their students. I encountered the consequences of this separation shortly after I began teaching high school mathematics. When I tried to show my students how the topics we were covering have real world, perceptual foundations, students typically demanded that I stop and just give them a formula that they could memorize. They had no interest in grasping the hierarchical chain that connects mathematical...