Origin of the Five Keys for UNDERSTANDING arithmetic and Achieving LASTING SUCCESS in the Subject

The five keys are based on the answers to the following question about a number fact like 5 + 8 = 13: What must students know to understand it, not just know that 13 is the right answer?

For one, students must know what the equal sign means. In computer science and on a calculator, it means “get the answer.” In mathematics, it means or “is the same as.”

Students must also know what the plus sign means and, for number facts like 13 – 8 = 5, 5 x 8 = 40, and 40 ÷ 8 = 5, what the minus, times, and divide signs mean, thus the ASMD (addition, subtraction, multiplication, division) key to get to the heart of the operations, that they are the same as occur widely in daily living.

The numerals, themselves, must be understood. The numeral 5, for example, stands for “fiveness”: the quantity that collections like 5 toys, 5 books, 5 ducks, 5 flowers, 5 boys and girls, … have in common. In each case, the quantity is determined by counting, thus the use of Tap & Tally and its system of touching and counting on the numerals to give them meaning and, by counting forward, backward, and skip counting on the “touch points” of the numerals, to enable students to the number facts instead of guessing for the ones they don't remember or aren't sure of.

The numeral 13, the answer to 5 + 8, also requires understanding. Why is it written with a 1 in front of a 3? Why isn’t there a special symbol for it like there is for 5 and 8? More generally, how is it that no more than 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are needed to express any quantity, no matter how big?

The answer lies in the nature of base 10 numeration, that the value of a digit in a numeral is the product of the digit and a power of 10 based on the digits “place” or position in the numeral. For 13, since the 1 is in the “tens” place, the overall value of 13 is 1x10+3. For a bigger numeral like 4,307, it would be 4x1000 + 3x100 + 0x10 + 7. Thus the to learn that the 1 in front of the 3 in 13 means that ten “the same” were traded for one of the next bigger thing.

The fifth or fractions key is needed to make sense out of 5/12 + 8/12 = 13/12. Unless one can literally picture what is going on, the answer is counterintuitive. Why isn’t it 13/24? Working with , like fraction circles, answers the question emphatically.

Put the five keys to the test. Pick up any elementary school math book and skim through the sections on how to add, subtract, multiply, and divide whole numbers, fractions, and decimals. Ignoring the technical jargon, like addends, products, least common denominators, …, none of which needs to be known to DO THE MATH, ask the question that led to the five keys: “What needs to be known to understand the math? What you will see is that all that changes in the books from grade to grade is the size of the numbers, not what needs to be understood. What needs to be understood for 23,478,049 + 8,906,382 is no different than for 5 + 8.