From our parents and teachers, we learn that, when performing a mathematical or arithmetical task, the result is a right answer or a wrong answer to a question posed by an adult. We learn that our answers result from a process and that there is only one process we can use to produce them. We learn that every question has an answer. We are pleased when our work produces a right answer and pleases the adult who asked the question and we are distressed when our work produces a wrong answer and displeases that adult. We conclude as young children that Arithmetic is rigorous, deductive, computational and useful. As our parents, teachers, and peers reinforce this conclusion throughout the first twelve years of school, we internalize and preserve it. It becomes bedrock to our view of the world and the places of things, events, and actions in it.
Neither History, Fiction, Poetry, Music, Physics, Chemistry, Biology nor, well, any subject is presented exclusively in this way. Sure, each field has its jargon to learn, its principles to apply and its recipes to follow. But, we learn in every field other than Math that authors embed their stories in context and tell them in several ways, teachers employ many devices and draw from many resources in their pedagogy. They tell the stories of their subjects as fields of inquiry that have evolved with civilization and like civilizations. They battle about the course of history, the nature and limits of fiction, music or poetry, or the proper use of its elements (character, plot, narrative voice, rhyme, meter, tone, key, scale, theme) the fundamental laws of matter and energy, the material composition of things or the building blocks of living things. Our teachers and parents demonstrate and encourage experimentation and discovery in every area—except Arithmetic and Mathematics.
During our lives, some of us find by chance a few corners of Arithmetic that are fun: Rubik’s Cube, Sudoku, magic squares, and ciphers among them. Others among us discover that Mathematics is quite different from Arithmetic and attempt to cross the divide between them. But, crossing this divide is arduous; everyone who has attempted this crossing knows this. Many of us decide that it’s too arduous, reverse course, and return to the safety and certainty of the Arithmetic and Mathematics we know, to the Island of Conclusions. The remaining few travelers complete the crossing and seek roles for themselves in this new land. What do they find there? Why is this crossing so arduous?
They find a vast network of communities of people, each of whom inhabits some area in an open, limitless landscape that we can characterize as an evolving field of inquiry (a field of evolving inquiry, perhaps). Some communities inhabit developed areas; some inhabit territories on the frontiers of Mathematics and others inhabit undeveloped territories beyond such frontiers. Each generation of inhabitants, called Mathematicians, explores, settles and develops land beyond its frontiers and discovers new relationships, new networks, within the developed territories. This story is the story of Mathematics; Arithmetic and school Maths are chapters in it.
We make the crossing to Mathematics, the field of inquiry, from Arithmetic, the body of knowledge, arduous by failing to prepare our students for it. If they need to cross an ocean, they must build the boats, learn to sail them and learn to swim. If they need to cross a desert, they must build the wagons or find and train the camels, and learn to navigate by the night sky, the position of the Sun, or by compass. But, we the communities of educators and parents, have failed to provide our students and children the boats, wagons, camels, or compasses or to show them how to build, train or use them. We’ve given them boats; they need ships. We’ve taught them how to use a calculator for Arithmetic and to replicate algorithms for Algebra, Geometry, Trigonometry and Calculus to find unique answers to manufactured problems. But, that is not enough. They need to learn how to recognize patterns and describe them with precision. They need to learn how to experiment with mathematical objects to discover patterns in their behavior. They need to learn to associate freely among concepts that appear similar or disparate, seeking connections they haven’t discovered, yet. For, although Mathematics is rigorous, deductive, computational and useful, it is, like any other subject, experimental, inductive and inferential. By framing it as the former alone, we hide the latter aspect of its nature.
Nearly all students learn to calculate, deduce and apply arithmetical and mathematical tools readily enough to get them through the days of their lives. But, the mathematics required for daily living is even more elementary than the reading required for it. Barcodes and computers total our groceries purchase and we use a plastic card to pay; no computation is required. Most daily math tasks are counting tasks. Calculators, hand-held or online, can calculate mortgage payments and their elements and other tasks that require more than counting. Students learn enough math for their daily living before they complete the seventh year of school. Demonstrations of and practice at applications of Algebra, Geometry, Trigonometry, Matrices, Vectors or Calculus convince our students that these tools are useful, indeed. But, such demonstrations and practice don’t convince them that they will ever use those tools. Mathematics’ usefulness in the world fails to motivate students to learn it after they’ve completed the sixth year. For the first three or four years, they needed only the approval of adults to motivate learning it; its usefulness didn’t matter to them. Once they’ve learned enough of the math they need to use daily, they lose interest in learning any further uses. Teaching them, in subsequent years, for example, how to use techniques in solving systems of linear equations in linear programming applications does not suffice to impel them to learn either how to solve systems of linear equations or how to apply those techniques in the world. The answer, “Because it’s useful,” to the question, “why do we have to learn this stuff?” is true, but it fails to stimulate students’ interest and impel them to learn such techniques and their applications. Only a few students will use them, and those students are unaware of their futures and don’t know, yet, that they will. WYSIATI (What You See Is What There Is) dominates their perceptions and their conceptions.
Mathematics is a body of knowledge, a compendium of results (truths) developed over at least seven millennia. Its method of verification is most rigorous of the methods of the sciences and it’s facts live longest of them. Logical deduction plays a unique role in its practice. Yet, Mathematics is more than a static body of knowledge. It’s an open, evolving field of inquiry into the nature of numbers and their relationship to the world. Its practitioners, Mathematicians, pursue answers to questions that arise in their endeavor to locate and understand the fundamental elements of the mathematical universe and explain their existence and behavior because such questions interest and challenge them and because they feel themselves continue to grow and learn, which they like to do. A Mathematician enjoys feelings similar to the feeling an author feels when he’s written a satisfying paragraph or an effective stanza or to a musician when he’s executed a passage “perfectly” or his performance jells with an ensemble.
Importance and applicability are reasons necessary to the pursuit of mathematical knowledge, and they are reasons necessary to the pursuit of any and all other knowledge. But, they are insufficient. In this regard, too, Mathematics is like all other fields of inquiry. We pursue our inquiries into them because we enjoy the process, and our success in discovering and answering questions amplifies this enjoyment a thousand times. The pursuit of mathematical knowledge is interesting, gratifying and creative. These factors impel practitioners to practice. These factors will impel students to learn to practice it, too.