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Beginning with this issue, a new column, "On a Clear Day," will be-gin. The column is written by Doris Glass, a nom de plume for a retiredscience/mathematics professor, and uses a looking glass to view the sci-ence/mathematics world.On a clear day, rise and look around you; there’s a world worth com-

menting on, and comment they do. Who? The lay press, the experts�of-ten misquoted or misunderstood, mind you, but comment they do."On a Clear Day" seeks to close the gap between science and mathe-

matics in an informal, narrative style, and to encourage critical thinkingabout what others are saying. Questions and observations are invited bythe author, in care of School Science and Mathematics.

On a Clear Day

Doris Glass

Sir John TennielArtist

Through the Looking Glassby Lewis Carroll

On a clear day Columbus discoveredAmerica. Not America U.S.A.: justa little island about six by ten milesthat he named San Salvador. Hehad about 150 San’s and 60 Santa’sto choose from, but October 12 be-longed to Salvador. You can find iton maps as Watling’s Island, 74°30 ’ East Longitude, 24° 00’ NorthLatitude. From Wailing’s Castleyou can see Christopher’s Monu-ment�on a clear day.

Is history, like science and mathe-matics, a mix of discovery, inven-tion, and memory? It seems Colum-bus knew about Hipparchus, thewell in Egypt, and his calculation ofthe earth’s circumference of 25,000miles. How far did Columbus hopeto go? On August 3rd, 1492, hesailed from Palos, Spain (37° 00’North Latitude, 7° 00’ EastLongitude) commanding the SantaMaria, Pinta, Nina, and 88 men.His logbook shows that he intended

School Science and MathematicsVolume 84 (6) October 1984

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to sail due west. Having studied astronomy, he could have sailed that far.Latitude was easy. It was longitude that was hard, because of the lack ofan accurate chronometer.On October 7th Columbus changed course, and headed south of west.

He never saw America, U.S.A. What if he had sailed due west? He mighthave arrived at Cape Henry, Virginia, where John Smith landed 115years later, at the entrance to Chesapeake Bay at 37 ° 00’ North Latitude,76° 00/ West Longitude!What is the point of this brief history lesson? Simply this. Are we at a

point of decision that, like Columbus’s, will affect the future in general,and the future of school science and mathematics in particular? Will theavailability of computers open up a new world? What lies beyond thehorizon?The purpose of this column is to explore. Dr. Irwin R. Whiteman,

computer software consultant, Encino, California says:

If it has its way, the California Board of Education will require all students,starting with the class of 1987, to take three years of English and socialstudies, two years of math and science, one year of fine arts or foreign lan-guage and some computer study�amount still to be decided. The new stand-ards, which local school boards will be urged to adopt, go far beyond whatmost California school districts required. For example, Los Angeles now callsfor only 1 Vi years of math and one year of science, with no requirements forcomputer study.

In the opinion of the state board, the right time for students to learn aboutcomputers is in high school. The earlier courses should stay as they are withyoungsters learning how to solve problems using paper and pencil. If the dis-tricts go along, it will again be the same old story of too little and too late.The moment of truth will come when students find out just how many jobopenings are only for those who can solve problems using computers.What the school board fails to recognize is that the problems of this ap-

proach are rooted in the fundamentals. The early grades present a false pic-ture. The student learns that arithmetic is slow, time consuming, inherentlyunreliable, a challenge to thought and concentration. True enough when wehad no alternative to hand calculation. But not true today when arithmeticproblems can be solved quickly and efficiently without mistakes.

It is wrong to continue the dated practice learning by rote, the endless daysof memorization and drill, working with solitary digits in every combination,the oral and written drills, over and over, again and again. It’s a travesty.The ages six through nine are among the most impressionable and precious.

How can we continue to squander them, developing skills and reflexes thatcan never match what a machine does faster and more reliably? The cast of ayoung mind lasts a life time; there can be no excuse for molding it into anarithmetical robot. The first years of schooling should not be directed to rote,but to discovery, the unfettered exploration of the unknown, the pitfalls andrewards of trial and error, the exercise of intellect.The devotion to hand calculation is an anachronism. At best it’s a waste of

time for those who quickly polish their skills; and it’s a disaster for those who


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don’t. Even when the humbler sets up solutions correctly, the numbers aremeaningless when the second time around doesn’t match the first, and whenthe third time doesn’t match the second. Just when the mind should be free toconcentrate on problem solving�as if this wasn’t challenge enough�it re-mains preoccupied with the simple sums. Only if the students develop anarithmetical proficiency can they go on to learn about problem solving. Ifthey don’t, they can’t. That is the harsh reality of the traditional classroom.The arithmetic of the first grade should be what the school board has pro-

posed for 10th-graders; pick up your calculator, enter the numbers and pressthe keys. The classroom would be simple and straightforward and everyonewould get a grade of A. The kids would love working with numbers; it wouldbe fun. Only after the youngsters had successfully tackled arithmetic with acalculator should they be introduced to the figures on paper.What is important is that the traditional sequence of teaching arithmetic be

reversed, the philosophy of how to teach it changed.That’s the way it could be and should be. But it won’t, not if the Board of

Education continues to have its way. And not that it stands alone, for it haslots of support from the PTA. Both groups are consumed by the fear thatchildren who learn arithmetic the easy way will not learn the hard way�thatwere it left to the kids, they would never learn the rules; that forever after,when faced with 1 + 1, 2 + 2, or 22 + 22, they will be slaves to the machine.

In the teaching of the three basics�reading, writing and arithmetic�therehas always been this confusion only about the fundamentals of arith-metic: that it can only be taught by rote. Yet there has never been this confu-sion about the teaching of reading, writing and its counterpart�spoken lan-guage. It has always been rather obvious that long before entering kinder-garten, most children have been talking in sentences, distinguishing betweennominative and objective, keeping verbs in agreement with their subjectsmost of the time, and using a large number of grammatical principles withoutknowing the technical names of any of them. All without conscious memori-zation, without structured drill.How can anyone believe that the child faced with 1+1 will forever be en-

slaved to a calculator and the touch of its keys? Anyone who does makes agrievous mistake. Once the youngsters learn that for certain classes of prob-lems they can move faster without the calculator than with it, they will.No one suggests that children who come to school with a good grasp of lan-

guage shouldn’t be taught the rules of grammar. Nor does anyone suggestthat children who learn to add, subtract, multiply and divide using a calcu-lator shouldn’t be taught arithmetic. What I do say is that it’s a matter ofwhen. Problem-solving should be practiced before, not after, learning therules of arithmetic.The answer to our educational dilemma is not, as the recommendations

would have it, just to add a semester of computer study to the high school,and students will not have to experience any surprises, nor will they have torethink all that was learned about problem solving.

What do you think?

Doris Glass
