In mathematics, the key to solving any problem or proving any proposition is applying current knowledge in such a way that the answer or proposition is demonstrated. Applying the known to understand the unknown is in fact the most fundamental of all scientific and mathematical principles. However, it is possible to learn to solve problems or demonstrate propositions without a more general possession of this essential skill. Through teaching by explanation—primarily through derivation and proof—it is possible to gradually introduce and then enhance this ability in students of mathematics. Proper exposure to and explanation of logical proof, combined with a clear exposition of the languages of mathematics and logic, will allow students to develop this intellectual faculty to its fullest extent. This power—the ability to apply the known to generate the unknown—is essential in creating not only capable students of mathematics but also educated men and women skilled in all aspects of life.
The literal meaning of the word science, from the Latin scientia, is knowledge. Webster’s dictionary defines science: "Knowledge of principles and causes; ascertained truth of facts. … The mathematical and physical sciences are called the exact sciences." All of science is concerned with the establishment of truth by the analysis of the facts. A scientific theory is an explanation of the facts based on known scientific principles. A scientific law is a mathematical equation describing the behavior of some scientific phenomena. A mathematical theorem is a proposition whose logical certainty has been derived from first postulates. In all these cases, the ends of science (theories, laws, and theorems) are all based upon and generated by truth: either accepted postulates or the observed facts. The ability to logically combine known facts to discover the unknown is the hallmark of a good scientist.
Mathematics, in fact, may be called the most unusual of the sciences in respect to its methods. It has clearly been shown that logically consistent systems exist based upon false postulates. The classic examples of this phenomenon are the non-Euclidean geometries. The foremost requirement of mathematics is not the ultimate truth of its system but its logical consistency. Thus to derive new truth in mathematics it is necessary to logically synthesize multiple separate statements: equations, postulates, or theorems. This is the mathematical form of applying the known to generate understanding of the unknown.
Every math problem, therefore, is a case of discovering the unknown by applying the known. This can be as simple as an addition problem, involving only the definition of the "+" symbol, or as complex as a proof of Fermat's Last Theorem, involving techniques and propositions from all different branches of mathematics. However, it is demonstrably possible to learn how to do a problem without developing the general skill of applying known principles to generate new knowledge. A student who learns to add numbers may flounder when given the definition of multiplication, despite having all the tools at his command to learn how to multiply on his own. In the same manner, a student may have a firm grasp of how to solve limit problems and operate on Riemann sums but find him or herself wholly unable to use these abilities to calculate the value of a definite integral. We therefore come to the one doom of all mathematics students: it is not enough to learn to solve the problems one is faced with—one must develop the ability to solve all problems by the logical synthesis of present knowledge and principles into a new, consistent answer.
Vital to this skill are the language of mathematics and a grasp of the meaning of logic. It is through these two components—precise symbology and logical demonstration—that mathematics synthesizes all of its principles into propositions and new answers. Deficiency in either of these areas will clearly eliminate any possible aptitude for mathematical study and render a student unable to keep pace with an advanced mathematics class in which some propositions must be solved without the guiding hand of the instructor. Hence, it is unquestionably required that all students must rapidly learn to understand the language and symbols of mathematics (including such terms as 'there exists', 'is an element of', and so forth) and gain a full and fundamental grasp of the principles of logic. Without these faculties they will be rendered completely unable to participate in any kind of mathematical discussion.
The present trend in mathematics education is teaching by exploration—students with the aid of powerful technology explore the possibilities of a mathematical environment and discover new concepts and relationships, with or without the help of lecture. This approach is widely touted as effective and helpful for students, and preliminary studies (and many anecdotal reports) indicate that this method is able to teach the concepts of mathematics (especially at the middle and high school levels) very quickly.
However, this method has a fundamental flaw: exploration of mathematics using technology gives little attention to the rigorous derivation of the concepts it teaches. The "why" of mathematical relationships cannot be explored by technology, but only by actual derivation from prior principles. The understanding of students of "exploratory mathematics" is therefore deficient. While they may learn quickly and be able to apply their knowledge to solve specific problems, they are unable to derive the laws they use from prior knowledge—a fundamental aspect of mathematical proficiency.
In a deeper sense, this lack of derivation and proof is damaging to the skills of the student as a whole. Without exposure to the language and symbols used to express mathematics precisely, and without experience using and understanding the principles of universal logic, students are ill prepared for future mathematics, science, and problem-solving. This is the true reason that a generation of exploration-educated students is frightening: they lack the sufficient training, experience, and understanding of logic and derivation to survive in a difficult and rigorous class in mathematics or the sciences, and this same lack of skill hurts their general mental faculties immensely. Abraham Lincoln was not a mathematician, but nevertheless learned to demonstrate all of the propositions of Euclid, because he knew that such ability trained the mind in the ways of correct thought. These abilities—logical faculties—played an evident part in the arguments that allowed Lincoln to win the presidency and develop a sound strategy for leading the Union through the Civil War.
To correct this major deficiency in educational methods, I propose a solution totally dichotomous to the prevailing doctrines of educational thought and "exploratory mathematics." A teacher must teach mathematics with little to no aid from technology and minimal "exploration". Technology, while powerful and useful to those trained in its application, is able to only display the mathematical relationship being taught—not the principles underlying its derivation and function. Teachers should slowly build mathematical concepts from original principles, introducing few concepts that cannot be directly derived from what is already known. This exposure to logic, the symbology of mathematics, and the occasional requirement of proof on the part of the student will gradually increase the student's aptitude for mathematical reasoning. Proof must not be arbitrarily introduced in high school geometry and then forgotten until linear algebra or basic analysis classes. Logical derivation of mathematical concepts must be present in all aspects of mathematics education, and its absence must always be noted and explained. (For example, it is not possible to prove that 'f(x) = e^(x^2)' has no elementary antiderivative until the theorems of Galois are known. This can be stated in a calculus class with the clear notation that the proof is outside the scope of the course.)
This methodology will result in expanded mental and logical capacity for students and enhanced understanding of mathematical concepts and principles. Furthermore, it will generate an ability to reason, synthesize knowledge, and perform research that will benefit any person for the rest of their life, whether or not they work in the fields of science. The ability to produce such abilities in students simply by altering the methods of mathematics education cannot be overlooked. Changing the course of the teaching of math will not only generate more competent students of mathematics but also improve the skills of mind that are so important to an educated person in general.
All of science rests upon the derivation of new conclusions from known facts or accepted principles. Mathematics is no exception—every proposition of mathematics must be suitably demonstrated by rigorous logic until it is accepted without possibility of doubt. The ability to solve problems by this method is vital to all students of mathematics. Without the ability to synthesize present knowledge into a valid answer, students miss the keystone of all mathematical study. Exploratory mathematics, while able to generate great knowledge in a short time, leaves students dangerously deficient in mathematical understanding and logical faculty. In contrast, teaching by explanation through proof enhances those very abilities that are so essential to both a student of science and any educated person. We must change the course of mathematical education to ensure the future of our children as skilled members of academic discussion and general educated discourse.
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