Fair and Equal in Political Mathematics


          Every history, poem, novel or other social commentary is written against a background of shared values, often values so common that the authors are unaware they are an essential part of even the most ordinary statements being made. Most literature from an earlier cen­tury, even the literature of one's own culture, will often startle the reader with this realization.                                      

          A hundred years ago it would have been unremarkable to hear or say, in Tennessee perhaps, or Michigan, the phrase "Spoken like a white man" intended as an expression of praise for someone's honesty. Similar­ly, "He's a Christian gentleman" was mere praise for that person's moral rec­titude.  Today's citizen looks on these phrases dif­ferently, often even doubting that they could have once been said in all innocent sin­cerity. Only those educated in the history and literature of that earlier time, hence educated more than is usual in today's schools, will escape making unwarranted judgments on the intent of such phrases as used in the past.


          For another example, familiar to those of us who have been studying the progress of the American thought police of the present day, Mark Twain's character Nigger Jim in Huckleberry Finn bears a mere name sufficient to bar the use of that book in many a school today. I need not belabor the explana­tion of the common as­sumptions that, in the minds of the ignorant, have generated such a reaction. Nor is the matter one of ignorance alone, for there is political capital to be mined by opportunists who knowing­ly employ the mistaken assumptions at issue.


          When it comes to the present time, it is usually difficult, some­times impossible, to recognize what there is in our present language, or in our social ‑ ‑ or even scien­tific ‑‑ as­sumptions, that will sound equally strange or maybe even evil to our grand­children's children. Much of our language is rooted in meta­phor -- "dead metaphors", as Fowler puts it -- whose meta­phorical origins are so forgotten that we no longer (again, unless we are scholars in language) recognize the assumptions they embody. These assumptions, if called to our attention by an interest group, are often painful enough to cause us to change our language, though sometimes mere ignorance, or the weak­ness of the potential interest group, permits their continuance.


          Ety­mology is not the only source of currently unnoticed as­sump­tions.  Some are philosophical and quite openly debated, but with a history long enough to have con­vinced most people that the debate is essentially over. In this country it is sufficient to say that we are a "democracy" to in­dicate that we value justice and freedom, though democracy does not of itself guaran­tee either, as the condemnation of Socrates attests. It might be that a hun­dred or two hundred years from now, democracy as we now know it will be looked on by our descen­dants as a quaintly provincial prejudice, American civic pride as repre­sented by the word "demo­cratic" by then appearing as nar­row‑­minded, though forgivable by the educated, as we today regard the 19th Century "Christian gentleman."


          That "democracy" will in fact be supplanted (two hun­dred years hence) by some other notion meta­phorically denoting justice, freedom or e­quality I cannot predict; this is all by way of specu­lation and illustrative example. It might be that justice, freedom and equality themselves will become regarded as out­moded vestiges of a shameful past, having been replaced by ideas of apparently greater virtue, towards which, like sincere 19th Century Protestant Christianity, the older notions were a halting prelude. Or, the words themselves might remain with their present virtuous connotation but with altogether different meanings. George Orwell's 1984 offers some suggestions in this direction.     


          Today there are two words related to justice that I con­sider problematic in this regard: "fairness" and "equality", and reading some literature apparently remote from either philosophy or poetry has sudden­ly called to my attention what might well be a genuine example of a present‑day assumption that not long in the future will, I hope, have become extinct. The literature in which this example occurs repeatedly is that of elemen­tary school mathematics education.


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          The report Adding It Up, published by the U. S. Na­tional Aca­demy of Sciences ((c) 2001, all rights reserved), can be found on the Web at <http://www.nap.edu>. It is subtitled Helping Children Learn Mathema­tics.


          The Mathematics Learning Study Committee, which wrote the report, was headed by Jeremy Kilpatrick, one of America's leading figures in mathe­matics education. The com­mittee com­prised mathema­ticians as well as professors of mathe­matics education and some others, all as excellent of their sort as are usually found for the purpose, and who cannot be considered ig­norant, narrow‑minded or even thought­less in their ideas and mode of expres­sion. I regard Adding It Up a good report, by the way, or mostly good, but even so it contains a few lines ex­hibiting the phenomenon of unconscious assumptions as men­tioned above, and in a way that may very well lead persons less learned than the authors to errors of a more serious sort.


          On page 7 of the Executive Summary of Adding It Up there is a section headed Developing Profi­ciency with Rational Numbers, whose second paragraph reads as fol­lows:


          Students' informal notions of partitioning, sharing, and mea­suring provide a starting point for building the concept of rational num­ber. Young chil­dren appreciate the idea of "fair shares," and they can use that understanding to partition quan­tities into equal parts. In some ways, sharing can play the role for rational numbers that counting does for whole numbers.


          Even from a strictly peda­gogical point of view I suggest that "equality" is understan­dable and teachable to small children without any reference whatever to "fairness", so that while the suggestion given in the quoted paragraph might be correct, in that some children might have been taught some­where that equality is fairness, so that the analogy has classroom value, it is by no means a necessary adjunct to lessons in fractions, or equality in partitioning.  Unfortunately the iden­tification of "equal" and "fair" is common enough in academe to have es­caped the notice of the Kilpatrick committee entirely, leading them to generally false doctrine as well as -- were the doctrine true -- an unneces­sary suggestion in pedagogy.                      


          But just as the man who praised his neighbor (say) with the words "Chris­tian gentleman" would never have thought, in 19th Century Boston, that his words were related to some mur­derous pogrom in equally Chris­tian Poland at that very moment, the mathematicians who wrote that para­graph about "fair shares" surely never saw it as connected with some serious misunder­stan­ding of mathematics. For while one cannot accuse the Kil­patrick commit­tee of equating the notion of "equal shares" and "fair shares" ‑‑ indeed, they subtly avoid pressing this e­quivalence as their own by putting "fair shares" in quota­tion marks ‑‑ they have in their pedagogical suggestion used a truly mischievous notion as the starting point for a mathematical lesson they really are not posing as a moral issue. Yet the subli­minal lesson might very well be reinfor­cing a problematic moral judgment of serious political consequence.                                          


          All mathematics begins in experience, and the assumption made by those writing this report was that this experience, in the case of young children, perhaps learned at home when fighting with siblings, has already taught them the equivalence of "fair" and "equal". What the Kil­patrick committee was trying to do was to make use of this confusion to teach what mathem­atical equality (in partitions) actually is.  Nothing more.  Yet the effect must be to rein­force a mistaken moral judgment.    


          My own recollection is that in my own child­hood fairness was nothing like equality. My older brother got the bigger piece of pie, and the bigger shirt; that was fair. If I had brought that notion of fairness with me when I went to school, and had my textbooks been written in accordance with the implied moralities of the Kilpatrick committee report, I would never have under­stood fractions. Or, though I doubt it, my schools might have taught me communism, in terms of which Frac­tions a la Kil­patrick would have been a piece of cake.


Ralph A. Raimi

July 13, 2003