To understand current Math teaching practice, IWY (I-do, We-do, You-do), we need to recognize at least three myths about Mathematics and Mathematics education—stories about what it is, whether and how it differs from other fields of inquiry, why we need to learn it, and how we learn it:
- Myth 1: Mathematics is unique among all fields of inquiry in its structure and practice. Mathematics is deductive and rigorous—and nothing else;
- Myth 2: Logical rigor and deductive method—and nothing else—must frame Mathematics pedagogy;
- Myth 3: Mathematics is a tool for use in the construction of other things such as buildings, rockets, theories and submarines—and is nothing more.
Math is different, right? There’s always a unique, right answer to every problem. Math is rigorous: you can always verify whether your answer is correct or incorrect with absolute certainty. When you read a topic section in a Math textbook or an academic paper, you encounter definitions, first, then theorems and their proofs. The theorems use the definitions to assert some claims about the world. These theorems follow by strict rules of formal logic from other theorems that were proven previously. A machine could, in principle, prove the theorems in your textbook. New research results are always published in essays in academic journals and rarely included (as new results) in textbooks. Although there is little or no exposition in an academic paper, there is almost always some exposition in a textbook, usually pictures, diagrams or graphs of a situation, or examples of algorithms, or all of them. The pictures, diagrams or graphs are heuristic devices; they motivate and depict the concept you’re supposed to learn to help you retain them for use. The examples are heuristics, too. They show you how to use the algorithms you learned to solve problems posed in the book, that is, how to do the homework problems. By telling the story of Mathematics only in this way in books, classrooms, and lecture halls, textbook authors, publishers, professors, and teachers present an incomplete picture of it. They create the myths that: Mathematics consists of definitions, theorems, proofs, and problems; theorems are deduced rigorously from other theorems by strict, formal rules of logic; and these theorems are deployed as algorithms to solve problems—and nothing else.
This story of Mathematics became the dominant narrative of Mathematics as the result of cognitive processes developed to promote our survival as a species over millions of years of evolution by trial and error. WYSIATI combines with representation, cognitive ease, and repetition to embed this myth in our memories and enable its instant, subconscious recall. WYSIATI is an acronym for What You See is All There Is. It is a slogan for the fact that the associative machine—our minds operating in “current awareness only” mode (which is 99% of the time)—dominates our awareness at any and all given moments; we are “hardwired” by evolution to jump to conclusions based on limited information. If the only story about Mathematics that you’ve been told is that it is rigorous, deductive, and computational, you will jump to the conclusion that it is only rigorous, deductive, and computational. This jump occurs early in school, in the first grade when we learn to add and before we’ve learned to reflect on and filter our immediate experience. Jumping to the Island of Conclusions is easy. But, you can’t jump back and there are no bridges or boats; you must swim back to the mainland, which is hard, so you are unlikely to try. Soon, we develop a stereotype of Mathematics that we use to represent it in all of our thoughts and emotions about it. This stereotype is reinforced by the repetition of IWY pedagogy over the first twelve years of school. It becomes the easy representation that we invoke subconsciously whenever we think about Mathematics or perform Mathematical tasks. By the end of first grade, we use this stereotype to represent Math throughout the school years and for the remainder of our lives, unless and until we learn another story and stereotype with which to replace it. Go on, try it! Try to think of Mathematics in some other way. How did you do?
The myth that Mathematics consists only of definitions, theorems, proofs, and problems (applications) frames our pedagogy and thereby limits our practice to stocking our student’s inventory of facts and algorithms to use them. We employ IWY as the default pedagogy because we believe this myth. We learned this convenient, easy representation of Mathematics from our professors, teachers, and textbooks, and we use it in our classrooms and lecture halls. It “worked” for us; it should work for everyone, shouldn’t it? We want our children, our students, to succeed, don’t we? After all, success is good; it reinforces what they’ve learned. Practice makes perfect. Homework is the opportunity to practice and, thereby, perfect and make permanent what they’ve learned. Solve the problems, achieve success, feel good about what you’ve done and who you are. Of course, we use heuristics to motivate topics. Thus, we use the rectangle to illustrate the commutative property: A rectangle’s area is the product of its length and width; the order of multiplication doesn’t matter (32 = 8 x 4 = 4 x 8). Or, we use a pair of scissors to illustrate the Hinge Theorem: the length of the side of a triangle is proportional to the measure of the angle opposite it; in a triangle, the side opposite a 60º angle is longer than the side opposite a 30º angle (√3 times longer). The hinge of the scissors represents the vertex of the relevant angle and the opening between the tips of the blades represents the side opposite the angle. As you open the scissors, the angle between the blades increases and the distance between their tips increases; as you close them, the distance between the tips shrinks as the angle between the blades decreases. We intend such heuristics to point to the path to understanding and retention, but they don’t.
Perhaps the most surprising and wonderful fact about Mathematics is its fruitful application to nearly every aspect of the world we inhabit. Its applicability is integral to the prevailing story of Mathematics. Applicability would, of course, be integral to any story of Mathematics. But, its role in this story is to play the sole reason to learn and do Mathematics. We learn Mathematics because it’s useful. Its value lies in its instrumentality. In at least one widely used high school textbook, at the beginning of each section there is a short list: what the section is about, what the student is going to learn how to do, and why the student should learn it. The third item in this list is, invariably, “It’s useful.” And, naturally, the section includes examples of its use in the world. Thus, we learn algorithms to perform such counting tasks as totaling the cost of a basket of groceries or the addition to your house or the size of your farm, or to perform such engineering tasks as determining the thrust required for a rocket of a given size to achieve escape velocity. We also learn algorithms to help us tell which algorithm(s) to apply to which circumstances. We call it “cookbook math” or “engineering math”. Our students—our children—do not meet “real” Mathematics until they’ve completed a twelve-to-fourteen-year apprenticeship that fails utterly to prepare them for this meeting.
So, Mathematics is, according to these three myths, in fact, and indeed, a body of knowledge consisting of definitions, theorems, their proofs and the algorithms that we use to apply those proofs to solve problems. Its value and, therefore, our motivation to learn it lie in is instrumentality. Together, these three myths comprise the meta-myth of Mathematics.