Drilling Procedures v. Thinking with Euler

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The values of classical education differ from those of conventional teaching in the West. The difference in educational philosophy shows up in methods. In teaching math, people take various approaches, some of them more successful and some of them less so.

Take the example of algebra. People often rush through concepts. They are prone to believe that concepts, once ‘covered’, are now understood. It’s important, I think, to judge accurately how ready a child’s mind is to think abstractly. Some children’s brains are physically not there yet. In that case, it would be better to do without the abstractions of letters standing for unknown quantities. Without the ability to think that abstractly, students end up frustrated and unable to learn what the teacher hoped they would learn. Instead, if they cannot yet understand the abstractions, they can do further work on concrete numbers. If they are not yet ready to understand, let them not delude themselves with the false impression that they understand.

Many Christian teachers use Saxon textbooks for math. The Saxon books take the approach of repeatedly circling back to drill procedures. Drilling mere procedures, rather than going for a clear understanding of what is actually being done, is a weakness that makes Saxon distinctly unclassical in its approach. I have some experience with children who used Saxon, and this is what I saw: Rather than sit down calmly and think about what the problem actually was, they grasped at straws to repeat some familiar procedure. Whatever had happened, their minds were dulled to actual understanding.

When we compare Saxon’s approach to teaching with Leonhard Euler’s approach to teaching his Elements of Algebra, we see a world of difference; and Euler’s, I think, is by far the more classical. Unlike newer books, which are eager to have students solve æquations of which they have no actual understanding, Euler does not begin to treat of æquations at all until section 4 of part 1, long after even logarithms:

  1. Containing the Analysis of Determinate Quantities
    1. Of the different methods of calculating simple quantities
      1. Of mathematics in general
      2. Explanation of the signs + plus and − minus
      3. Of the multiplication of simple quantities
      4. Of the nature of whole numbers, or integers with respect to their factors
      5. Of the division of simple quantities
      6. Of the properties of integers, with respect to their divisors
      7. Of fractions in general
      8. Of the properties of fractions
      9. Of the addition and subtraction of fractions
      10. Of the multiplication and division of fractions
      11. Of square numbers
      12. Of square roots, and of irrational numbers resulting from them
      13. Of impossible, or imaginary quantitites, which arise from the same source
      14. Of cubic numbers
      15. Of cube roots, and of irrational numbers resulting from them
      16. Of powers in general
      17. Of the calculation of powers
      18. Of roots in relation to the powers in general
      19. Of the method of representing irrational numbers by fractional exponents
      20. Of the different methods of calculation, and of their mutual connexion
      21. Of logarithms in general
      22. Of the logarithmic tables that are now in use
      23. Of the method of expressing logarithms
    2. Of the different methods of calculating compound quantities
      1. Of the addition of compound quantities
      2. Of the subtraction of compound quantities
      3. Of the multiplication of compound quantities
      4. Of the division of compound quantities
      5. Of the resolution of fractions into infinite series
      6. Of the squares of compound quantities
      7. Of the extraction of roots applied to compound quantities
      8. Of the calculation of irrational quantities
      9. Of cubes, and of the extraction of cube roots
      10. Of the higher powers of compound quantities
      11. Of the transposition of the letters, on which the demonstration of the preceding rule is founded
      12. Of the expression of irrational powers by infinite series
      13. Of the resolution of negative powers
    3. Of ratios and proportions
      1. Of arithmetical ratio, or the difference between two numbers
      2. Of arithmetical proportion
      3. Of arithmetical progressions
      4. Of the summation of arithmetical progressions
      5. Of figurate, or polygonal numbers
      6. Of geometrial ratio
      7. Of the greatest common divisor of two given numbers
      8. Of geometrical proportions
      9. Observations on the rules of proportion and their utility
      10. Of compound relations
      11. Of geometrical progressions
      12. Of infinite decimal fractions
      13. Of the calculation of interest
    4. Of algebraic equations, and of the resolution of those equations
      1. Of the solution of problems in general
      2. Of the resolution of simple equations, or equations of the first degree
      3. Of the solution of questions relating to the preceding chapter
      4. Of the resolution of two or more equations of the same degree
      5. Of the resolution of pure quadratic equations
      6. Of the resolution of mixed equations of the second degree
      7. Of the extraction of the roots of polygonal numbers
      8. Of the extraction of square roots of binomials
      9. Of the nature of equations of the second degree
      10. Of pure equations of the third degree
      11. Of the resolution of complete equations of the third degree
      12. Of the Rule of Cardan, or that of Scipio Ferreo
      13. Of the resolution of equations of the fourth degree
      14. Of the Rule of Bombelli, for reducing the resolution of equations of the fourth degree to that of equations of the third degree
      15. Of a new method of resolving equations of the fourth degree
      16. Of the resolution of equations by approximation
  2. Containing the Analysis of Indeterminate Quantities

For brevity, I have omitted the chapter headings of part 2, ‘Containing the Analysis of Indeterminate Quantities’.

What Euler’s way demands is attention and careful thought, not mindless following of steps to perform numerical sorcery. For that, I appreciate him greatly, even if students have often to be compelled to think hard about what they believe they already understand. A calculator is able to take an input and give an accurate output; a human mind, however, is able to understand, and that is the difference between mastery and slavery in the mind.

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