In most socially oriented activities, one of the first topics to surface in discussion is one’s profession. When it is discovered that I have selected mathematics education, I continually receive a solemn stare of surprise, bewilderment, disgust, and empathy. Individuals outside and within the profession of mathematics misconstrue the nature of mathematics, the pedagogy of mathematics, and the scholarship of mathematics. A mathematics educator must have a well-informed understanding of mathematics based on academic, personal, and professional discovery.
The nature of mathematics is not defined; it involves posing questions with reliable conclusions. “Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality (Courant, 8).” As early as the age of two, children begin posing the question “Why?” Most often, parents and psychologists refer to this as a phase, but this is justification of the entire function of education – to satisfy an inquisitive mind and provide answers. Since medieval times, the “three R’s” have been emphasized in instructive curricula: Reading, wRiting, aRithmetic. Each discipline has a unique function. Reading is intended to explain knowledge that has previously been discovered. Writing is intended to endow knowledge that has been gained through inquiry-based learning. And arithmetic is intended to explore that which is known with the intent to find that which we do not possess. Arithmetic is a complex concept and defines itself as the process for finding a solution. “Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science (Courant, 8).” The nature of mathematics is the culmination of previous experience, knowledge, and the satisfaction of the infamous question: “Why?”
Mathematics appears to be a subject for the elite and simply a “hobby” for the intellectually gifted, but mathematics is beyond numbers, symbols, and operations. According to the National Council of Teachers of Mathematics (NCTM), mathematics is problem solving, communication, reasoning, and connection. (Fromboluti, 11) Mathematics is taught in schools to enhance these skills. Mathematics is problem solving and developing an understanding of procedures to attain an intended solution. Mathematics is developing skills of reasoning, which is used to develop a useful answer after intensive thought and deliberation. Mathematics is making connections to concepts, subjects, and matters of everyday life. Mathematics is communication, finding ways to express ideas through words, diagrams, pictures, and symbols with the intent of making their own thoughts clearer. “Classroom instruction, of any kind, is a system. It is made up of many individual elements that work together to create an environment for learning (Heibert, 7).” A mathematics classroom should be a place where: inquisitive learning takes place, questions are developed and solutions are established, applications are practical and realistic, a community of learners is established, classroom discourse is facilitated, students learn from one another, engagement is facilitated, learning is driven by interest, learning from mistakes is encouraged, the primary assessment is non-traditional, the role of the teacher is shaped by the goal of facilitating conceptual understanding (Heibert, 8), understanding is not the privilege of high achieving students (Heibert, 11). One of the hardest roles of a mathematics educator is posing questions. Teachers are responsible for facilitating thinking from students and as such, leading questions are imperative. It is important to take the following considerations into account before asking a question: “encourage reasoned guessing – not savage guessing, establish the norm that students’ responses should include a rationale, always be aware of who is doing the thinking – the teacher or the student (Grouws, 137).” The pedagogy of mathematics is being able to foster knowledge acquisition in students through engaging, innovative, and rigorous task creation, implementation, and completion.
For scholars and layman alike, it is not philosophy but active experience in mathematics itself that alone can answer the question: What is mathematics? This active experience is essential for knowledge to be created or invented by a student. Learning is an active, not passive construction of ideas and beliefs about reality. These ideas and beliefs should be integrated into students existing knowledge base, making the subject more meaningful. (Clements, 1) Learning is seen as a culture in which students are involved not only in discovery and invention, but also in a social discourse involving explanation, negotiation, sharing, and evaluation. The nature of mathematics should be seen as making sense of reality, not learning a set of procedures with little or no connection to reality. (Clements, 2) Another way to actively involve students in learning mathematical skills and content is through the use of manipulatives. Don’t be misled to believe that “mathematical knowledge will automatically arise from the use of manipulatives (Ball, 18).” The use of manipulatives is growing in popularity among American educators and the benefits that such instructional tools offer are undeniable. Manipulatives provide teachers with a tool that can be used to explore, represent, and communicate mathematical ideas. Manipulatives can allow students to see mathematical truths, which leads to conceptual understanding. Another way to actively involve students in learning and knowledge acquisition is though project-based-learning. Allowing students to work on a real world problem or a project that will serve a purpose beyond the educational benefits is an ever-growing method of instruction. When students can see the “fruits of their labor,” they are more likely to invest themselves, set personal goals and actively work to achieve knowledge acquisition. “Formal mathematics is like spelling and grammar – a matter of the correct application of local rules. Meaningful mathematics is like journalism – it tells an interesting story…The best mathematics is like literature – it brings a story to life before your eyes and involves you in it, intellectually and emotionally (Stewart, 2).” The scholarship of mathematics is like the most intriguing, thought-provoking, engaging story where the student is the main character.
Mathematics and all other sciences are seriously threatened when it is implied that mathematics is nothing but “a system of conclusions drawn from definitions and postulates that must be consistent, but may be created by the free will of the mathematician (Courant, 11).” If this were an accurate description of mathematics, it would be a game with definitions, rules, and syllogisms, without a unified motive or goal. As an instructor of mathematics, it is my goal that students see mathematics for what it is: a system of fundamental laws and theories that define and allow us to predict the world within which we live. Stan Gudder, professor of mathematics at the University of Denver, says it best, “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”
Ball, Deborah L. “Magical Hopes: Manipulatives and the Reform of Math Education.” American Educator: The Professional Journal of the American Federation of Teachers 16.2 (Sum 1992): 14+. Print.
Clements, D. H. & Battista, M. T. (1990). Constructivist Learning and Teaching. Arithmetic Teacher, 38(1), 34-35. ©1990 by the National Council of Teachers of Mathematics.
Courant, Richard, Herbert Robbins, and Ian Stewart. “What Is Mathematics?” Foreword. What Is Mathematics?: an Elementary Approach to Ideas and Methods. New York: Oxford UP, 1996. 8+. Print.
Fromboluti, Carol Sue., Natalie Rinck, Diane Magarity, and Barbara Gibson. Early Childhood Where Learning Begins : Mathematics : Mathematical Activities for Parents and Their 2- to 5-years Old Children. [Washington, DC]: U.S. Dept. of Education, Office of Educational Research and Improvement, National Institute on Early Childhood Development and Education, 1999. Print.
Grouws, Douglas A. (2004). A Vision for the Teacher’s Role in Teaching Mathematics Through a Problem-Solving Approach. In R. Charles (Ed.), Teaching Mathematics Through Problem Solving: It’s About Learning Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Hiebert, James. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann, 1997. 1-27. Print.
Stewart, Ian. “Preface to the Second Edition.” Preface. What Is Mathematics?: an Elementary Approach to Ideas and Methods. New York: Oxford UP, 1996. 1-4. Print.