Math teacher blogs are rife with frets over the mantra "When are we ever going to use this?" and its implied corollary "Why are we learning this?" That should lead teachers to examine the inverse question: "Why are we teaching this?"
Speaking at the broadest level, there are two basic reasons for teaching a piece of information: It is inherently important to know in and of itself, and some aspect in the learning of it represents a relevant cognitive process. It is generally assumed that graduates of a course on US History will know that Thomas Jefferson wrote the Declaration of Independence in 1776: This is knowledge per se. It is not taught to show how historians process information, it is taught as something that is inherently important to know. Meanwhile, in English class, students are carefully and critically reading a poem by Anne Sexton. It doesn't matter what poem it is. It doesn't even have to be Anne Sexton; it could just as easily be Sylvia Plath. While certain facts of Sexton's life might be shared in the process, the emphasis is on the process of looking at poetry; indeed, the facts about Sexton's life would be shared as part of the literary analysis process (Sexton was an American poet most active in the 1960s and known for confessional poems on depression... how does knowing this affect how we look at her poetic style?).
All content areas have both "facts to know" and "procedures to learn". Social studies tends to be heavy on the facts, and ELA heavy on the procedures, while science and mathematics sit somewhere in the middle.
Because mathematics is somewhere in the middle of the spectrum, it's important to reflect on what "facts" there are to be learned, and when we're teaching something for the point of learning a procedure or a "way of seeing". I feel, based on having reviewed several textbooks and talking to colleagues, that the emphasis in mathematics (Common Core aside) has become one of learning facts, with the learning of underlying processes taking a supporting role. There are obvious reasons for this: It is easier to test awareness of facts, for instance, than the test underlying cognitive processes. "Teaching to the test" too often relies on going through sample tests and teaching the specific information needed to answer the question successfully; when I do ACT/SAT tutoring, I focus on strategies instead of specific information, because someone with successful strategies will know how to attack a problem they've never seen before.
I have been thinking about the taxonomy of quadrilaterals often taught in high school geometry class: All squares are rectangles, rhombuses are parallelograms with all sides congruent, and so forth. I noticed that, while Larson Geometry (e.g.) has an entire chapter on the properties and taxonomy of quadrilaterals, the Common Core math standards barely mentions them, and does so most clearly as a THIRD GRADE standard: "Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories."
Let's look at the Common Core standard without the examples: "Understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category." Where could students use this information? It's useful in science, where animals can be categorized based on similarities and differences. What characteristics do humans have because we're human, what do we have because we're primates, or because we're mammals, or because we're warm-blooded? It's useful in computer programming, which in modern days is heavily reliant on objects which inherit properties from other items. It's useful any time someone deals with taxonomies: A car dealer, for instance, can help a customer compare sedans to each other, and to see the difference between a wagon and an SUV. The Common Core standard, as written, shows that an understanding of the taxonomy of quadrilaterals is useful because of the underlying process of categorization: It is not knowledge to be learned per se, but because it represents a cognitive "way of seeing".
But is that coming through to students, or even to teachers? There's nothing about "a rhombus is a parallelogram with four congruent sides" that immediately emphasizes the generalization of categorizing objects. It's a fact: It defines a rhombus. If we had to choose, do we want students to know a specific fact about a specific object, or do we want them to understand about categorization and property inheritance?
In a video on the Erdős Discrepancy Problem, which is all about series of pluses and minuses, Dr. James Grime comments, "So why do this? ... The real reason we do it is because it's a hard problem, and the methods we develop to solve this problem could then be used in other problems, perhaps more important problems." He admits the problem has limited appeal in and of itself; the point is, what can the process of solving it tell us about how to solve other problems?
I'm not denying that there are important facts to learn in mathematics. If my students leave Geometry without knowing the Pythagorean Theorem, for instance, I feel I haven't done my job. But I do think that, far too often, we math teachers focus too much on specific facts and not on the process of developing the facts, and how that process can then be applied to develop other knowledge.