Santa Barbara, Spring 1961
New Math

In the 1950's, being able to accurately add, subtract, multiply and divide was an important life skill.  The transistor effect had been discovered in 1947, and the integrated circuit was patented in 1959, but hand-held electronic calculators would not become inexpensively available for another 20 years.  Today, solar-powered calculators the size of a credit card can be purchased at the grocery store for less than a dollar -- one quarter the price of a loaf of bread -- but at the time, such a device was unimaginable.    In 1960, mechanical cash registers were common and there were mechanical adding machines (you punched the number you wanted to add and pulled a handle to cause the number to be mechanically stored, punched another number, pulled the handle again...).  These devices were big and EXPENSIVE, so they were used in business and not found in homes or schools, much less in anyone's pocket.   So everyone needed to learn to do arithmetic and do it well!

As a result, lot of the instructional time in grade school was spent memorizing tables used in numerical calculations.  In the first and second grades, the students memorized the addition and subtraction tables for the numbers 0 through 9.  In third grade, multiplication and division were introduced and the students memorized the tables for multiplying and dividing the numbers 1 through 5.  In fourth grade the tables were memorized for the numbers 6 through 9.  Fourth grade was also the level at which the students were first introduced to dreaded "long" division.  

In Mrs. Hill's fourth grade class, we did daily drills on "our tables."   Fridays were celebrated with a timed test where every student was presented with a gigantic list of fiendish problems and had to do as many as possible in ten minutes.  

For many students, fourth grade was also their first introduction to negative numbers. This came about because one wrong answer on the timed test meant that the credit for two right answers was deducted from the final score.  Failure to attain an acceptable grade on Friday meant a loss of precious recess time the following week and more drill.

However, the year that I was in fourth grade was the year that Washington Elementary School introduced New Math. 

The Wikipedia describes New Math as:

...a brief dramatic change in the way mathematics was taught in American grade schools during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost scientific education and mathematical skill in the population so that the supposed intellectual threat of the Soviet engineers, reputedly highly skilled mathematicians, could be met.

Two kids from our grade were picked to be trial new math students and I was one of them. Instead of having to memorize multiplication tables and do the dreaded timed drills, the two of us, the designated guinea pigs, followed a completely different curriculum from the rest of the class. 

They had math workbooks with long, long lists of boring multiplication and division problems. 

We had books that explained interesting things like how different number systems worked.  

They had timed tests. 

We got to do puzzles.

Social skills were not part of the New Math curriculum.   My math partner and I were little snots and publicly reveled in the fact that we got to play with numbers while the rest of the class was stuck learning their 6's, 7's, 8's and 9's times tables.  It is a marvel that we both survived to the end of the school year and that neither of us was murdered by our classmates.

I don't think that math was Mrs. Hill's forte -- she once startled the entire class by very humanly announcing after a Friday test that she really hated long division too and was very sorry it had ever been invented.  

New Math seemed VERY interesting and it all seemed to hang together logically, but I could not figure out exactly what it was for.   Mrs. Hill had no idea what to do with it either, so she just let the two of us work the way through the books on our own.   When we got to the end of the chapter, we'd do the test from the teacher's workbook and then go on to the next chapter or book.

It must have been very hard on her to be expected to explain exponents, algebra and logarithms to a couple of curious 10-year-olds.  I thought the idea of number systems with more or less than 10 integers was pretty cool. Furthermore, Pop had given me a slide rule when I was in third grade and taught me how to use it, so logrithms weren't entirely a mystery, but exponential expression was new to me and I didn't know any algebra.

I remember that while Mrs. Hill understood how to square a number, she was shaky on raising a number to any power larger than 2 and particularly flummoxed by the idea of a zeroth power.  In fact, I don't believe that she agreed that zero was really a number, it was "...just a placeholder."  And if she liked the number zero not at all, she actively disliked ∞ and didn't think it belonged in a math book.   She certainly let us know that she didn't believe the text when it stated that zero and ∞ behave very much the same when multiplied or divided.  I remember her complaining to us while checking our work: "Zero is nothing and infinity is everything. How can nothing and everything be alike?  This new book they've given us doesn't make any sense!"

She completely cratered when we got to the chapter on fractional exponents.  School books were never allowed to leave the building, but (probably hoping they would get lost) Mrs. Hill made a special exception for the new math books and when there were questions thereafter, I was allowed take the offending volume home for Pop to explain. 

New Math was intended to be self-paced, so our classroom set included not only the books for elementary students but also those intended for junior and senior high school.   No one stopped us and, given that the alternative was timed tests and long division, Marc and I worked our way through almost all of the books by the end of the year.

That year the school department also implemented a standardized exam to "uniformly evaluate every student's performance against objective criteria for graduation" or some similar scientific silliness.  The idea was that if a student passed at a higher level than their grade, that they would subsequently be exempted from doing the lower level work.  Marc and I were at least as shocked as Mrs. Hill when the scores came back and both of us had passed the supposed "exit exam" at the high school level and were to be declared "officially" to be done with math.  I remember that Mr. Dal'Armi, the principal, called us to his office and congratulated us and Mrs. Hill too, but I don't think she approved of it at all!

An unfortunate side effect of the affair was that by summer I could explain the theory and attributes of the operation of multiplication but could not reliably multiply or divide in the base 10 number system if any number larger than 5 was involved.   I paid for my fun that July when Auntie Gladys and Uncle Paul, who were horrified to discover this fact, proceeded to drill me on my times tables and long division skills all the way to Oregon and back.