When I became a Customs Agent in San Francisco, the amount of time I would spend in a car came as a surprise. Fortunately there was a great classical music station in the Bay Area, and I heard a lot of Telemann, Vivaldi and Corelli. But the mind wanders none the less when one is driving a long time and often. Somehow I began working simple little math puzzles in my head. I noticed that when I took two instances of a number, 8 for example, added 1 to one instance and took 1 from the other and multiplied the two together (7*9) the answer (63) was always 1 less than multiplying the two instances of the original number. More days of driving and playing with this in my head, with different offsets, that is adding and subtracting 2, then 3, I noticed that the differences 1,4,9,16 were the squares of the series, 1,2,3,4 and were the sums of the odd number series 1,3,5,7. Fascinating! After a long time of this, I then wondered “how would I express this in symbols?” So I began playing with that in the back of my mind, all the while navigating traffic and thinking of whatever investigation I was working on. When I finally had it, I was mortified, even there in the solitude of my car, that, after all that time and all that fascination, I had merely rediscovered one of the most elementary concepts of beginning algebra: the factor x(squared)-1 = (x+1)*(x-1), and, the more general form x(squared)-y(squared) = (x+y)*(x-y). Now why was this embarrassing, even though there was no one else around to witness this? Because I was a science student in high school, even tutoring college students who were having trouble with algebra and trigonometry, and I was a math major my sophomore year in college. I could derive the quadratic equation, integrate and differentiate, calculate the maxima and minima of curves, and calculate the area under a curve or the volume of a solid resulting from rotating a curve around the x axis. I had known for year, YEARS! that x(squared) -1 = (x+1)*(x-1) and x(squared)-y(squared) = (x+y)*(x-y). I had used those factors in solving equations hundreds of times. But I never knew what that meant in numbers; never knew it meant that if you took two instances of a number and took 1 from one instance and added 1 to the other and multiplied them together that the answer was 1 less than multiplying the original numbers together. That is, I knew something perfectly as an abstraction, but only as an abstraction. It has made me wonder ever since how much more I know as abstractions but never connect to the real world, such as ‘love your neighbor as yourself,” and it has made me try ever since to see beyond abstractions to the reality they represent.

Ron Hicks, Parish Verger, St. Alban’s Episcopal Church, Washington DC

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4 Responses to Abstraction

  1. Linda Wilson says:

    Well, as a mathematics educator, I never thought I would read an essay that would link the factoring of a quadratic with love of neighbor. Thank you, Ron, for making that connection!

  2. Elinor says:

    Again, wonderful What do you make of Pascal’s Triangle??? Elinor constable

  3. Christian says:

    I thought — “Where is he going with this?” Good gut punch at the end.

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