Math and the Common Core Standards
(from a letter to AFT)
Dear American Federation of Teachers,
Your efforts to bring about national standards in education over the past few years have been admirable; and the Common Core Standards are admittedly better than the standards I have seen here in California. They are better, though, in the way that Ptolemy’s astronomical model was an improvement on the Aristotelian one. Rather than reforming flaws on the fundamental level of existing state standards, renovations have been made to the latticework. Again, this is better than what we had, but not enough to merit the accolades bestowed upon it by AFT.
Your most recent editorial apologist for the greatness of Common Core Standards, Hung-Hsi Wu, penned an article proving my point in the most recent (Fall 2011) issue of American Educator. The article, entitled, “Phoenix Rising: Bringing the Common Core State Mathematics Standards to Life,” offers up and defends the new standards as a panacea to the pedagogy of mathematics. Wu thoroughly demonstrates how the same problems inherent in the old standards have been carried over to the new.
His most explicit declaration of the fact lies in the insert (pp. 8-9), “The Fundamental Principles of Mathematics,” in which he lays bare the a priori beliefs which drive the mathematics standards, both old and new. He states these assumptions as follows:
1. Every concept is precisely defined, and definitions furnish the basis for logical deductions.
2. Mathematical statements are precise. At any one moment, it is clear what is known and what is not known.
3. Every assertion can be backed by logical reasoning.
4. Mathematics is coherent; it is a tapestry in which all the concepts are logically woven to form a single piece.
5. Mathematics is goal oriented, and every concept or skill has a purpose.
According to Wu, many teachers and textbooks compromise their students’ learning by violating these principles. I would argue, however, that the development of the mind of each student is compromised by adhering to these principles. Let us look at each, in turn.
The first principle betrays a blind faith in the axioms and postulates of analytic logic and mathematics. Such a logico-deductive dogmatism can be damaging to the student, as indicated by the following scenario. An instructor offers up the definition of a prime number to her class. A precocious student asks why it is called prime, and why it has come to be defined in that way. The teacher replies that the term and definition were established long ago, and that the important thing is to remember what it means so it can be used in mathematical operations. The message to the student: just believe in it now, because it is an important player in the game of math. That student’s curiosity on the matter was effectively destroyed. No mention was made of the remarkable work done by the Pythagoreans and Euclid on primes, or the alternative “definitions” of primes furnished by Gauss. The idea of establishing, as if ex nihilo, concisely expressed statements of mathematical principles runs contrary both to the nature of mathematical discoveries and the mind. Gödel’s critique of Russell’s attempt to formulate a complete formal deductive system equally applies here. The idea of a perfect math “game” with rules and definitions is an illusion; and this is, in part, why students feel so utterly frustrated with math.
Wu’s second principle continues in the same vein. Yes, it is important for students to be precise in mathematics; and yes, there must be a clear sense of logic in mathematical operations and proofs. Yet, to say that it is clear at any moment what is known and not known, tells us more about Wu and the state of mathematics today than about mathematics itself. Wu has fully invested in a purely analytic system of mathematics, and cannot bear the thought of traversing its boundaries. Whereas, in the history of mathematics, all crucial discoveries have been made precisely due to a sense of confusion between known and unknown. Again referring to Gauss, his first method in discovering the Fundamental Theorem of Algebra did away with all formerly known boundaries of powers and their roots, and it was within that realm of paradox and temporary confusion that his new frontier was found. To present math as only precision and clarity is to present its perceived effects divorced from the causes which generate them. Again, we can see why the minds of most secondary math students experience a sense of morbidity, and are thus so repulsed by the subject.
The third and fourth principles of Wu carry the same flaw of a dogmatic faith in Russell’s dream of a complete system. The active principles generating mathematical concepts are ignored, as well as the required mental actions (i.e., processes of discovery) of those who discovered them. In such a formal system, math is presented to students much as a taxidermized dog is presented to a bereaved dog owner. As students advance in their math education, they feel an ever increasing sense of loss. They know not from whence this feeling comes, but I can assure you it is from too much time spent working with a corpse.
In the fifth principle, Wu temporarily breaks from his profession of faith in all things dead, and invokes the idea of pragmatism. The idea is that math is useful, and that students who realize the usefulness of it will enjoy and understand it all the more. It is a “skill” that will help them solve problems and succeed in life. At first one may be surprised to see this fifth principle—different from all the rest. It makes perfect sense, however, when one considers how the first four principles have annihilated everything that is interesting and beautiful in mathematics. What is left but to convince students that, though they despise what they are learning, it will help them to be happier later in life?
Those are the principles by which Wu, and the Common Core Mathematics Standards, operate. Let us look at them in practice. Wu provides two main examples to demonstrate the superiority of the system.
First we are shown a convoluted method for adding fractions. Instead of finding common denominators, converting the fractions accordingly, and then adding them (as in most state standards), Wu and the new standards force the student’s mind down onto a one-dimensional linear system—the number line. Though his explanation fills over a page, it merely consists of dividing up the number line like a ruler, and then showing how a multitude of fractions can express the same interval. By dividing up the intervals in two ways, by the two denominators in a given problem, the student can see which fractional units are equivalent and add accordingly. He argues that students will like this and excel at it because they are used to “combining things.” Since students have been “combining things” all through grade school, they should be kept bound within the limits of elementary combination so as not to confuse them. The Common Core method does indeed make it easier to produce students who will properly carry out the operation of adding fractions, and consequently score better on tests. What, though, does such a method do for the development of the thought processes of the student? Are we allowed to consider such a question, or is the only matter of concern here that the students be properly trained to master the prescribed content? In this system, genuine exploration into the nature of fractions is forbidden. Confusion is anathema. And, just as with the old standards, the concept of discovery simply does not exist.
Wu’s second example of an improvement in the new standards, is an explanation for multiplying negative integers. Instead of demonstrating to students how to multiply (-1) x (-1) = 1, Wu suggests giving them (-1)(-1) + (-1) = (-1)(-1) + 1(-1) = [(-1) + 1](-1) = 0(-1) = 0. The second equation, he claims, has greater benefit to the student because it allows them to “’compute’ in the usual sense of arithmetic.” Again, he is insisting that students be kept bound to the primitive axiomatic system with which they have entered the classroom. By keeping them in that system, and then building within it a structure of syllogisms, students are comfortable with the content and can prove such on assessments. Is this what math has come to? Is this what teaching has come to—the mere producing of results regardless of underlying mental action?
When Riemann delivered his habilitation dissertation to the faculty of Göttingen, he declared that the very foundation of mathematics was shrouded in darkness. Mathematicians had placed self-imposed limits on the nature of space itself, affecting everything from mensuration to quantity. In our system of education today, we have done the same, with both the old standards and the new. I welcome any improvement in the field, but at the same time ask that we do not settle for anything less than the best possible. The new Core Standards are part and parcel of the old. The formulation of them is a tragic example of missed opportunity. There was an opportunity to create a national system of education exemplifying an early American tradition—that expressed by the Massachusetts Bay colony. There was an idea, once, of taking the best of the great things the intellectual endeavors of humankind, everywhere, had to offer, and transmitting them, full and alive, to progeny through an organized system of education. That idea has become lost. We have settled for the status quo, and for the past five to ten decades have been tossing around lifeless shells of “content,” calling it education. However, like Wu, I also have faith. It is not faith in a static framework of logic and symbols; it is a faith in progress. I see the current state of education to be a mere stumbling block on the long road to excellence. Riemann ended his lecture by asking that “progress . . . not be obstructed by the prejudices of tradition.” I ask the same, and hope that the new Common Core Standards are just a very temporary solution.
(October 7, 2011)