RightStart and Why Learn Math
Joan A. Cotter, Ph.D.
Mathematics is one of the greatest tools we humans have inherited. Math is given to us both as a dedicated brain area and as a legacy from society. Galileo understood the importance of math when hundreds of years ago he said: “The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics.” Math enables us to understand the cosmos and the atom; to build bridges, tunnels, and skyscrapers; and to determine through statistics which treatment is most likely to cure an illness.
Also, mathematics is necessary for the arts; some examples are musical scales, drawing in perspective, and repeating patterns. Mathematics has even been defined as the science of patterns. A person becomes a mathematician, not because they enjoy arithmetic, but because they see the beauty in mathematics. And math is not necessarily utilitarian. One of its 200 branches is recreational mathematics, with magic squares as an example.
The field of mathematics is expanding exponentially, doubling about every seven years. Many times a mathematical innovation initially seems to have no practical use. Take the case of imaginary numbers, which involves square roots. We know the square root of 1 is 1 since 1 × 1 is 1. But what is the square root of –1? It can’t be 1 because 1 × 1 is 1 and it can’t be –1 because –1 × –1 is not –1, but 1. In the 17th century, a few mathematicians thought the whole idea was so absurd that they referred to the square roots of negative numbers as imaginary numbers, labeled either i or j. Today imaginary numbers have an important role in AC (alternating current) calculations.
The ancient Egyptians used math, including the Pythagorean theorem, to survey their land annually following the Nile River flooding. They also needed math for their constructions and for astronomy. They were interested primarily in the practical applications of math.
The ancient Greeks, on the other hand, approached mathematics, especially geometry, from a philosophical perspective and devised a rigorous deductive approach for proving theorems, or propositions. Each step in proving a theorem is based on one of three rationales: axioms, universally accepted truths; definitions; or previously proven theorems. Subjecting theorems to diligent explanations at every step of a proof has become the cornerstone today of all mathematics, not only geometry. However, we no longer insist, as did the Greeks, that all geometric constructions be done with only a straightedge and compass; drawing board equipment and computer programs are a welcome addition.
Euclid compiled and wrote the 13 books of Elements around 300 B.C. Amazingly, Euclid’s Elements has undergone about a thousand editions, including translations into numerous languages, and remained a textbook for over 2000 years. No other textbook can rival that!
In the Middle Ages, the trivium composed of grammar, rhetoric, and logic, was the introductory topics studied. Coming next was the quadrivium, which started with Plato, and included arithmetic (number), geometry (number in space), music (number in time), and astronomy (number in space and time). Every educated person was familiar with Euclid’s Elements. Abraham Lincoln carried a copy in his saddlebags, which he studied at night by lamplight.
Most high school students study Euclidean geometry, where they are introduced to proofs. A good math program prepares students for this next chapter in their mathematical education by requiring learners to justify their thinking whenever practical. Uncertain about the concept of proof, students sometimes fail to realize that a proven theorem applies in all cases. It must.
Incidentally, mathematicians do not use the two-column structure, where statements are in the left column and reasons are in the right column. This arrangement is practiced only in introductory high school geometry classes. Mathematicians use ordinary prose to discuss the validity of each step.
Just as we don’t limit our reading vocabulary to words having a practical use, we must not limit mathematical topics to what we judge our children will need in the future. Even Bonnie, age 13, understood this. After learning about the Golden Ratio she commented, “It’s just one of those things in life that makes you feel satisfied to know.”
Despite the human brain being programmed for mathematics from infancy and the long history of humans being involved with mathematics, almost half of the people in the U.S. have a fear of numbers. They have no shame admitting having low math skills. Sadly, those harboring math anxiety are less likely to become our coders, engineers, researchers, and so forth. Until recently, immigrants filled this gap. Today a real shortage exists.
Of course, the real question is why are our children developing math anxiety. Preschoolers invariably do enjoy math, so the problem occurs later. Researchers have found some answers including: using flash cards and timed tests, having a teacher who harbors a dislike for math, believing the myth that a “math mind” is needed, and treating math as though it’s a collection of rules to be practiced and memorized.
RightStart Mathematics was written to develop future generations who are confident in their mathematical abilities. This program emphasizes visualization skills over counting or rote memory, games over flash card drill, place value to the thousands over memorizing counting words to 100, problem solving with thinking over key words, linear fractions over circles, and geometrical constructions over learning definitions.
Down through the centuries, people believed learning Euclid’s math produced logical thinking; that assumption has not been proven. But, we do know that early mastery of number sense predicts better reading and math proficiency years later.
And yes, a person who was taught math poorly can relearn and finally understand as they teach and become one of those who say, “I wish I had learned math this way.”