Studying mathematics

The power of teaching is seldom very effective, except in those happy and fortunate cases when it is almost superfluous. Edward Gibbon, Miscellaneous Works, 5 vols., 2nd ed., London, J. Murray, 1814


For every non-trivial topic, it is never possible to give those who are completely ignorant of the subject matter of simple and eusary answers in a few words, quickly clarifying their doubts. But it is possible to glimpse the key concepts, the main ideas of each scientific subject by chasing three metaphors (Irving Adler ^[1]):

  • LANDSCAPE: go through it with a bird’s eye view of a vast landscape of mountains to climb and well-hidden valleys
  • GEMMA: liken it to a rough gemstone that gradually reveals in patient workmanship the brilliance and colors of a thousand facets
  • TRAILER: telling it like a movie trailer, which through a series of images entices you to see the whole movie.

Things should be explained as simply as possible. But not too simply, otherwise they only create the illusion to have understood and so many misunderstandings. Instead at the opposite studying everything from the beginning in too much depth and rigor is tantamount to falling down an endless rabbit hole, without the certainty of hitting bottom, as Alice succeeds, but only in Wonderland. This is especially true of mathematics, in which the great power of abstraction, generality and rigor, should not let people forget that when tackling a new problem, intuition is more important, approximations, special examples, practical applications.


Brilliant secondary school and early college students, and the motivated and talented self-taught, people who have not had the opportunity to take formal courses at the level university on modern science, or have abandoned them, but continue to read and inform themselves throughout their lives, interested in the language of mathematics, they should:

tD flowchart
a1([ Look for motivation in its <br> applications to physics, science, technology, and the real world. ])
a2([ Studying various examples of applications of a concept, <br> to understand the need for a general definition, <br> encompassing all the various cases ])
a3([ See broadly the historical development of the subject, <br> in its social and cultural context ])
a4([ Learning the main concepts, informal definitions <br> and the most important results, <br> ignoring demonstrations and exercises at first ])
a5([ Learning to program and carry out experiments <br> and computer simulations, <br> to link concepts to their applications ])
a6([ Do lots of exercises with pen and paper, gymnastics for the mind, <br> or, if problems solvable only numerically, <br> continue with computer experiments ])
a7([ Deepen the subject by studying rigorous definitions, <br> axioms and detailed demonstrations of major theorems ])
a8([ Writing summaries and notes for lectures, <br> as if you had to teach everything to someone else ])
a1 ==> A2 ==> A3 ==> A4 ==> A5 ==> A6 ==> A7 ==> A8

Mathematics is a part of physics. Physics is a science experimental, part of the natural sciences. Mathematics is that part of physics where experiences are cheap. Jacobi’s identity (which forces the heights of a triangle to meet at one point) is an experimental fact in the same way in to which the earth is round (i.e., homeomorphic to a sphere). But it can be discovery at less expense. In the mid-20th century, attempts were made to divide physics and mathematics.The consequences proved catastrophic. Entire generations of mathematicians were formed without knowing half of the their science and, of course, in total ignorance of any other science. they began by teaching their ugly scholastic pseudomathematics to their students, and then to schoolchildren (forgetting Hardy’s warning that for bad math does not there is permanent place under the sun). Since school mathematics cut off from physics is not suitable neither for teaching nor for application to any other science, the result has been universal hatred of mathematicians-both by the poor pupils (some of whom have since become ministers) than of users

Vladimir I. Arnold , On the Teaching of Mathematics, Russian Mathematical Surveys, vol.53, no.1, 1998, pp.229-234, translated in CRITICAL POINTS No. 3 (MAY 2000)


Arnold’s provocation should be contextualized in the controversy against the teaching of mathematics inspired by the Bourbaki group, harshly criticized for two reasons:

  1. Separation of Mathematics and Physics: Arnold criticizes the tendency to separate mathematics from the other sciences, as physics, a consequence of the bourbakist movement, leading to a generation of mathematicians who do not understand the other disciplines and teach a “school math” lacking practical applications.

  2. Axiomatic and Formal Approach: The Bourbaki group was known for its strictly axiomatic approach and formal to mathematics, which Arnold considers too abstract and distant from intuition and practical application. This approach, according to Arnold, leads to a teaching that emphasized logical derivation of incomprehensible concepts, rather than explaining their meaning and usefulness.

In summary, Arnold criticizes bourbakism because he believed it made mathematics too abstract and separated from the other sciences, losing sight of the importance of intuition and practical application.

References:


Chinese teachers adopt four basic principles: modeling, mastery, variation, and mathematical structures.

  • Modeling involves the simultaneous use of visual, symbolic and procedural representations; for example, in teaching fractions, they combine visual models of area, line and sets with symbolic calculations.
  • Mastery ensures that each student develops deep knowledge before progressing to new content.
  • Variation, through examples and counterexamples, stimulates critical thinking and the ability to generalize.
  • Attention to mathematical structures leads the teacher to build concepts progressively, encouraging abstraction and generalization through constant dialogue and interaction with students

References:

[1]: Irving Adler, Ruth Adler, Peter Ruane - A New Look at Geometry, 1° ed. 1966, new ed. 2013.
Adler is well known for the 1952 U.S. Supreme Court case, Adler vs. Board of Education, by which an appeal was made against the dismissal from all schools of teachers suspected of progressive political views and left. It was the particularly dark period for democracy and justice of the McCarthyist witch hunt, and in the U.S. several liberticidal laws and ordinances were enacted. Adler lost the first time in court, and was then first suspended from his mathematics professorship, and later dismissed. But he continued the battle for teaching freedom, trying again and again, until fifteen years later the Supreme Court agreed with him and reinstated all the teachers who had been kicked out. In the meantime, he had a lot of free time and wrote more than 50 volumes including mathematics textbooks and popular books.