First Grade With Concepts: How I Learned to Multiply Without Memorizing Tables.

This post will ground a discussion of the existential implications of the differences between an education proper to humans - a conceptual education - and the pseudo-education foisted upon Americans by Pragmatist "educators," and increasingly spread by them to the rest of the world. When Ayn Rand discussed the issue in The Comprachicos, the Pragmatists were still in the initial stages of taking over. Today, many adult Americans don't even know that there is any workable alternative, to the Pragmatist dogma that education consists of (1) learning skills and facts by practice and rote, without conceptual understanding; and (2) of testing "educational achievement" by examinations during which the student has no time to think and no time to apply concepts - and the teacher has no choice but to "teach to the test" by imposing rote memorization on the student, and prohibiting conceptual thought lest the student think, and "waste time," on the eventual test.

A properly human, conceptual education has only the most limited contact with memorization. Most things that can be memorized, can be as fluently learned by repeated reasoning from first principles. I can multiply as fast as, and probably faster than, an adult who did memorize the multiplication tables in first grade. After reasoning out some specific result from first principles two or three times, the result will be remembered and retrieved faster, than if it had been memorized. And, having been understood, that result will be usable in solving problems beyond the grasp of the memorizer. (There are contexts where memorization is necessary - a surgeon cannot take the time to think about evolutionary anatomy while operating - but such contexts are rare.)

I had the enormous good fortune to have learned first grade arithmetic with the Łukasiewicz curriculum. Jan Łukasiewicz, an Aristotelian philosopher of mathematics best known for Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, had been, in 1919, independent Poland's first Minister of Education, and created its distinctly Aristotelian K-12 mathematics curriculum. This is what I remember of the sequence of ideas in first-grade Arithmetic:

1. Digital (finger) counting; cardinal and ordinal numbers; addition.

2. Doubling; odd and even numbers; halving of even numbers.

3. (Single digit) subtraction; zero and negative numbers; equations.

4. (Single digit) multiplication (beyond doubling) as repeated addition.

5. Exponentiation as repeated multiplication; powers of 10.

6. Number bases; place-order (Arabic) notation; the carry; multi-digit operations.

7. Multiplication shortcuts.

8. Modular (Clock) Arithmetic; telling time; clock arithmetic (time) operations.

Multiplication shortcuts took the place of memorized multiplication tables. All the shortcuts had a derivation from previously integrated concepts, so that I had a conceptual understanding of what I was doing:

x 2, AKA "doubling:" Add the starting number to itself.

x 10. Move the digits one place (base 10!) to the left and put a zero at the end (in the ones' place - the number of ones, after multiplication by the base, is zero.)

x 5. 5 is half of 10, so first multiply by 10, and then halve the result.

x 9. Multiply by 10 and subtract the original number from the result.

x 4. 4 is the second power of 2, so double twice.

x 3. 3 is (2 + 1,) so double and then add the original number again.

x 6. Multiply by 5 and add another instance of the original number.

x 8. Double thrice.

x 7. Multiply by 5 and add twice the original.

In a comparative test after the first grade, we (unlike a class of memorizers) would have understood what we were doing, although we would have been slower, than memorizers just recalling what they learned by rote. By the end of the second grade, we would have had enough accumulated practice to be equally fast - and incomparably superior in understanding what was going on, and in being able to figure out new calculation methods for new contexts.

The essential advantage of learning conceptually is that what one learns makes sense. And that makes multiplication, and all of Arithmetic, and all that comes after, natural and easy. There wasn't a single pupil in my first-grade class who could have thought, "I am not good at math."