Mastering Math

The study of the material world reaps great benefits for the material welfare of mankind: food, fabrics, energy-saving devices, heating and cooling, transportation,luxuries of every kind, cures for diseases and pain. Science and technology have transformed our world into one of unimaginable wealth, comfort, and blessings for which we must give thanks and gratitude to all those who have gone before us and those who continue to  work with nature to make our lives better. To study science and math is to learn about the spirit and ingenuity of man—to learn about the physical world that is his home and how it can be used for his well-being.

Mathematics is the language of science and the indispensable tool for the study of the natural world. The key to success in all sciences, especially chemistry and physics, is a good math education. Because mathematics, music, and Latin are the three universal languages, they are given a special emphasis in classical education and are required of all students every year. As with Latin and music, we aspire to have the very best  mathematics education possible. Languages develop and discipline the mind to a greater degree than other subjects. Mathematics, like all languages, is cumulative, rigorous, and demanding; it develops logical, accurate, and precise thinking habits.

Arithmetic is the art of counting and calculation; it does not require any higher-order thinking skills. There is nothing to understand about arithmetic—it is the memorization of math facts and algorithms. In the trivium model of learning, arithmetic is the focus of kindergarten through sixth grade. Like the alphabet and the Latin grammar, arithmetic is a finite subject, suitable and satisfying to the young child’s concrete mind. It is something over which the child can achieve mastery. There aren’t many things like that in school or in life. Next to the alphabet, arithmetic is the most useful tool students will ever possess. We tend to take it for granted, but it should not be rushed over in a hurry to get to higher mathematics. The goal is to master arithmetic with speed and accuracy so that students can do higher math when the time comes, without thinking about  arithmetic at all.

Additionally, arithmetic teaches accuracy, precision, and attention to detail. Each time the student checks his own work he learns the importance of quality work and doing a job right. These are priceless skills in character formation and in marketable work skills. Mathematics and Latin are the great enemies of the superficial, shallow, lazy, and inaccurate work that is the natural default mode of the young, and indeed of the human race. Mathematics, like Latin, teaches students to do thorough, honest, accurate work, which is a skill and habit of mind of inestimable worth, far above the content of the subjects themselves.

You cannot practice arithmetic too much in these younger years. It’s like piano. It’s not enough to know the notes. You have to hit them with speed and accuracy or no one is going to listen to you play. If you plan to play fluently, you must practice for at least an hour a day for many years. Forget composing your own music or playing Mozart—you are not there yet. You need a keyboard sense. And forget attempting algebra—first you need number sense and immediate recall. Games, drills, skip-counting, cypher drills, and challenge problems can help make arithmetic an enjoyable part of the school day.

After mastery of the arithmetic, students are then ready to embark on the study of mathematics, the science and philosophy of relationships. It is a large and varied subject that includes many topics, such as algebra, geometry, and calculus. Topics from the world of mathematics are suitable for students who have reached the age of abstract thinking in middle and high school.

Many modern textbooks do not observe the distinction between arithmetic and mathematics. This is one of the characteristics of progressive education: There is a disdain for basic skills and memorization and a tendency to introduce higher-order skills without laying the foundation to get there. Most students in modern education are pushed too fast in mathematics and fail to experience the kind of mastery that ensures continued success and enjoyment. They are victims of superficial and shallow coverage that emphasizes exposure to advanced concepts without mastery. In these modern textbooks you will see algebra topics included as early as kindergarten, which obscures the necessary arithmetic skills for each grade. The energy and effort of students are dissipated over a variety of topics, many of which are too abstract for students to understand. Insufficient time is given to basic skills and students do not experience the satisfaction of mastery learning, nor are they adequately prepared for higher math. You can read a book you only partially understand and it won’t hurt you, but attempting higher-level math concepts without a mastery of arithmetic is not only confusing, but harmful. Careful preparation is the watchword.

Mathematics is an exact language, one that is unrelenting and unforgiving. A failure to master basics is the cause of the glass ceiling that most students experience in their math education. A small weakness in the beginning, unless corrected, magnifies over time and eventually becomes an insurmountable obstacle to continued progress. Laying a solid foundation in mathematics requires overlearning, achieved by repetition to the point of automaticity.

It is important to get math right. Don’t be swayed by Johnny down the street who is doing algebra in third grade while your Johnny is “just” doing multiplication and division. Your Johnny is mastering the math he is learning. And don’t be taken in by the idea that computation skills aren’t important in the age of the calculator. There is this idea that since we have machines that can do multiplication and division and fractions for us we should not be spending time drilling math facts. But immediate recall of math facts is an absolute necessity in order to do higher-order thinking in math. Developing number sense through mastery is the only sure pathway to success in math in the upper school years.

I see as much confusion about math pedagogy as Latin, probably more. You have to spend the whole year on arithmetic in the lower grades. You cannot devote one unit of the year to computation and the rest of the year to abstract math concepts students aren’t ready for and that are hard to teach.

• Primary: The goal in kindergarten through second grade is to learn to print numbers and to learn addition and subtraction facts.

• Third: Third grade introduces multiplication and division, which students continue to work on until mastery.

• Fourth: In fourth grade we master long multiplication.

• Fifth: In fifth grade we master long division, the culmination of arithmetic skills. A long division problem is a long process with room for many mistakes, so students must have all their math facts cold.

• Sixth: In sixth grade we add mastery of fractions, decimals, and percents, which utilize the mastered arithmetic skills.

To succeed in any subject you must know what the priorities are. There are many things to do every day, but to be successful you must do the essential thing first, and you must do things in the right order. The essential thing in the grammar school years is arithmetic.