Mastering Math Helps Students Learn How to Learn
Download the original document (RTF)
"Thus my design is not here to teach the Method which everyone should follow in order to promote the good conduct of his Reason, but only to show in what manner I have endeavored to conduct my own. Those who set about giving precepts must esteem themselves more skillful than those to whom they advance them, and if they fall short in the smallest matter they must of course take the blame for it."
Rene Descartes, 1637
Discourse on the Method
At Plato's famous academy in Athens, an inscription over the door read: "Let no one enter here who is ignorant of geometry."
Plato, Archimedes, Aristotle, Pythagoras. To the ancient Greeks, no academic subject was more important than geometry. And among all math brains, Euclid was undeniably the master of all masters.
It is said that Euclid was teaching geometry in Alexandria, Egypt, and King Ptolemy heard of his amazing calculations and grew curious to observe a class. The pharaoh spent some time listening attentively and, mesmerized, asked Euclid to give him instructions. The teacher was flattered. Starting at the beginning, Euclid demonstrated some basic theorems. King Ptolemy followed the lesson in silence but after a while interrupted the exposition.
"This is fascinating," he said, "but as a pharaoh, I have many other duties to perform, and I have little time. Is there an easier road to the mastery of this subject?" Euclid gently replied: "I am sorry, sire, there is no royal road to mathematics." There are no short cuts or laborsaving methods to make math easy or always "fun." It takes time and practice and thinking and work. Math is both a skill and an art. It is a skill in that anyone who is willing to dedicate a bit of effort to its study should be able to maneuver in it competently. It is an art as there are some people for whom it seems to just come naturally.
Math is exact, abstract and logically structured. These are essential and characterizing properties of mathematics, which enable it, when properly taught, to make unique and indispensable contributions to the education of all youth. Adolescents need the experience of working in a subject where answers are exact and can be checked for consistency with known facts.
Adolescents have just discovered capabilities of abstract thought and proceed to exercise them without restraint. In the intellectual sphere, they have a tendency to become involved in abstract and theoretical matters. Adolescents construct elaborated political theories, invent complex romantic plots or develop plans for the complete reorganization of society. In the process of exploring their new abilities, adolescents sometimes lose touch with reality and believe that they can accomplish everything by thought alone. Mathematics presents them the most brilliant example of the extension of abstract reason without the aid of experience. Math builds confidence and perseverance that adolescents need to tackle tough intellectual challenges and to stick with an assignment until they find the right answer. Practicing this kind of mental accuracy teaches them to think critically in different situations, to identify the strengths and weaknesses of statements, to distinguish truth from nonsense, and to develop clear and steady understanding. Mathematics helps them refine the one indispensable ingredient for human endeavor: use of judgment.
Logically structured thinking enables the students to toss other people's arguments into tow piles: good ones and bad ones. Since logic is the language of rational thought, success in the study of math will yield the students a double benefit: they will not be likely to be misled by poor arguments of others and they will be likely to produce good arguments of their own.
In sixth grade, the gradual development of formal logic must begin in the curriculum and must be given appropriate emphasis. The middle-school years are ideal for developing the symbolic language along with a vocabulary that ought to include key word definitions such as and, or, not, then, set of numbers, element of and also definitions of the main group of numbers: natural, integers, rational, irrational and real numbers. Also, relations and functions should be formally defined and fully integrated into graphic representations of all trigonometric and algebraic equations. Not least, the logical reasoning confers the ability to formulate the contradictions of conjunctive, disjunctive and implicative statements when they reach high school.
As in any other discipline, capable teachers play a critical role in all this. The well-prepared teacher is skillful in a variety of approaches and in deciding what approach to use in specific classroom situations. The talented math teacher serves mostly as a director of learning, as an expositor who imparts knowledge and understanding by direct whole-class instruction. As a painter who chooses the right brush for a certain stroke, the math teacher chooses the proper technique for a given subject. Parents don't send their kids to school to learn from other kids.
By the way, parental expectations are the greatest determinator of a child's professional success later in life. If the students are academically viable, they will most likely be professionally credible. Parents should always communicate a high level of expectation regarding their children's academic abilities.
The most challenging task while teaching a math class is keeping the whole group focused. There are so many peripheral issues continually competing for attention, such as equity, anxiety, diversity, self-esteem and multiculturalism. All these are distinct subject matters. Effective math teachers try to teach the students how to think, not what to think. Efficient math teachers should also define a set of performance goals and expect these goals to be met in objective tests. Clear accountability and healthy competition prepare the students to cope with the competitive world that confronts them.
In these days, math students can count as well on unprecedented help: pyrotechnically illustrated books, graphing programmable calculators and compatible computer software. But a complete and legible classroom notebook is still their most reliable tool for success, and fine math students are usually successful in all other disciplines. They have learned how to read intensively for meaning, how to focus on a job and how to listen. And this might be the main reason to go to school: to learn how to learn.
At high school age, children are looking for their limits, and mental discipline contribute much to their sense of security and well-being, allowing for exploration of new concepts within the realm of a familiar routine and expected boundaries. Yet, some argue that even the most elementary fundamentals of logic needed for mathematical discussions are too difficult for secondary-school students. When these are omitted from the classroom instruction, a student who says that he understands the formula for the cosine of a difference generally means that he has memorized the formula and can apply it. Indeed, in mathematics, the axioms ought to be over-learned to a point where they are recalled effortlessly. But memorization alone is not enough. Relying on memory only serves a limited purpose: ultimately, the cumulative effects of these deficiencies overwhelm him and he leaves mathematics in frustration and despair. Many of them graduate with self-esteem but these students have not reaped any of the benefits that should be obtained from a properly taught course in school mathematics. This adds up to a mind-wasting form of child abuse.
The majesty of mathematics comes from the evidence of its demonstrations and the power of its reasoning. Blindness to this aesthetic element is widespread and can account for a feeling that math is dry as dust, as exciting as a phone book. Otherwise, mathematics possesses not only truth but also supreme beauty, and this element makes the subject live in a splendid and passionate manner, as no other creation of the human mind seems to do. Mathematics has a beauty remote and solemn, like that of a sculpture capable of a stern perfection such as only the greatest artists can show, sublimely pure, and without appeal to any part of our weaker nature.
Fernando Multedo has an undergraduate degree in industrial engineering from the Universidade Federal do Rio de Janeiro, Brazil, and holds a doctorate in education from Columbia University in New York. He is the founder of the New Mexico Academy for Sciences and Mathematic, in Santa Fe, NM, USA.
← Back to Multedo Campus