A Note on Mathematics and Curricula
EDUCATION, 2 Apr 2018
A study is going on comparing mathematics curricula in various countries. Interesting to see what they have picked up, and why. What follows is some reasoning about the what and the why.
This author was fed a high school curriculum rich on Euclid and Pythagoras in geometry, and on second degree equations Y=aX2+bX+c=0. There was a link: the geometry of a parabola. Solutions came as a formula and as the points where the parabola cut the X-axis. The parabola was hoisted up and down for two, one, or even no solutions: the X-axis was insufficient to accommodate solutions.
The teacher missed a chance to bring in the concept isomorphism. His colleague in geography talked about isotherms and isobars for the same temperature and pressure, preparing us for weather forecasts. But “same structure” is a deeper concept. Terrain and map. Attitude and behavior, at least as an hypothesis. There may be hatred with no action, and there may be cold-blooded murder. The teacher might have said that behind a2 + b2 = c2 and X = -b +/- &c there is a major way of thinking and learning: you learn map and van walk the terrain if the map is accurate, the isomorphism is solid. And vice versa.
I solved the last second degree equation in my life at the bac exam and have later seen none of that, nor of Pythagoras. But, isomorphism has accompanied by adult life as a mediator, triggered by asking myself “What does this conflict, this situation remind me of?” What was the solution in the other, could it translate into this one? Can “strong lady vs weak husband”, “strong Peru vs weak Ecuador high up in the Andes”, marriage and geopolitics on the rocks, translate into each other if only as an hypothesis, not as an a priori fact? They can, and more so the richer the isomorphism; very helpful method.
Why do millions complain that mathematics is not only difficult but also boring and out of this world? Maybe “isomorphism” would help.
Have a look at “mathematics”, “math” for short. Etymology often helps: Greek roots about “learning” point to math as basic. The word usually comes as plural, explicitly in French as les mathématiques. There may be several mathematics. Some “definitions” lists them, as geometry, arithmetics &c. But these are definitions in extension, like defining “Europe” by listing the states. Intellectually lazy when the real task is to get at the intention, or an essence, of both.
Mathematics is about abstractions, Europe about individualism. Mathematics is not about pairs but about what pairs have in common, “two-ness”, symbolized by ‘2’. Mathematics is not about squares but about what squares have in common, “squareness”, symbolized by a square. Moreover, these abstractions relate to each other in theorems, for instance about prime numbers; 2, 3, 5, 7, 11, 13, 17, 19, 23 &c.
Definitions establish new abstractions, proofs new theorems, and transcendence new mathematics for problems not accommodated by the old (like negative integers, fractions, roots, imaginary-complex numbers).
Mathematics takes on its own life, by invention and by discovery. Thus humanity has created something bigger than itself, on top of us, not vice versa. Like digital reality, also growing by being explored. Artificial intelligence, robots in general, are now lining up.
But this is not so new as it may look. Humans have created state systems making them slaves and victims of state competition and wars. And a feudalism with very many toiling and dying to benefit very few. And slave systems with humans being bred, bought and sold like cattle. Gender systems against women; generation systems against young, old. All more or less at the expense of nature being depleted and polluted. But, with longer and better lives in equity and harmony within reach.
Can mathematics be of any use, and if, what kind of mathematics?
The key would be isomorphism, “same structure”, between some social reality and some mathematics. Which ones? Learn from the past.
Newton and Galilei worked on mechanics, the movements of physical bodies subject to forces, Newton first with celestial, Galilei with terrestrial bodies, No jumps — Natura non facit saltus — deterministic. The mathematics of real numbers served them well. Differentials and integrals were defined. Mechanics was in the math box, that math box.
But social reality is often jumpy, discrete, not only continuous; and not easily caught in deterministic laws because of complexity, and because “laws” may become self-denying, not self-complying, prophecies. Not determinism but probabilistic, no straight lines, parabolas &c in X,Y diagrams, but scatters. Means, deviations, correlation coefficients are artifacts with no social counterparts; to be avoided. Medians make sense, modes even more, and additive indices often work.
Continuous vs Discrete and Deterministic vs Stochastic (process-oriented probabilistic) combined divide mathematics in four parts. Arithmetics is discrete and humans count, if only one, two, several. Algebra — from Arabic al-jabr, reunion of broken parts, the similar in the dissimilar — starts with sets and adds relations for structure; mirroring sets of actors related by “interaction” for structure.
Matrices mirror N-ary relations, as do graphs of points connected by unbroken/broken lines for positive/negative relations, both for ambiguity, unconnected for none. Matrices multiplied mirror two-step relations seen in graphs, easily learnt social science tools combining the geometric with arithmetic counting of associated numbers &c.
Four types of mathematics with “continuous-deterministic” adequate for celestial and terrestrial mechanics (if friction-free in a vacuum) and “discrete-stochastic” adequate for social phenomena like level of societal cohesion. Waiting to be stochastically enriched.
And the curriculum? A taste of all; with math mirroring the world and the world mirroring math. Start with graphs for the class.
Guaranteed outcome: nobody will find mathematics boring.
* This author’s 1956 thesis in mathematics at the University of Oslo, guided by this kind of thinking, was “Stochastic Relation Matrices”.
I am indebted to my teachers in geometry, arithmetics and algebra, professors Ingebrigt Johansen, Viggo Brun and Thoralf Skolem, also for their tolerance of my often deviant ways of approaching their fields.
Johan Galtung, a professor of peace studies, dr hc mult, is founder of TRANSCEND International and rector of TRANSCEND Peace University. Prof. Galtung has published more than 1500 articles and book chapters, over 500 Editorials for TRANSCEND Media Service, and more than 170 books on peace and related issues, of which more than 40 have been translated to other languages, including 50 Years-100 Peace and Conflict Perspectives published by TRANSCEND University Press. More information about Prof. Galtung and all of his publications can be found at transcend.org/galtung.
This article originally appeared on Transcend Media Service (TMS) on 2 Apr 2018.
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