Students should be trained in reasoning — they should not be unmoored intellectually as the ones you can see in my last post. (That’s sad…and infuriating.) Those students aren’t being trained in the first principles and objectivity they need for proper self-discipline, for being independent and being their own person, for understanding of the world and acting in it (“Nature, to be commanded, must be obeyed” — quote from Francis Bacon (1561–1626). Novum Organum, bk. 1, aph. 129 (1620).).
Students should be trained to reason, to think in principles — as they are in my classes, and as can be seen in a brief, sketchy note I sent to a student. I emailed the note, a short reminder about some of the things we had discussed in a tutoring session (getting the student (a high school senior) ready for a standardized math test), the same day as the session. Here is the note:
Remember to look for first principles in math — as in history, science, biology and life.
We frequently use facts like a number times one equals the number; any number times zero is zero; zero in the denominator of a fraction is not a defined fraction; and zero in the numerator of a fraction is zero.
1. So when we need to add fractions, we must multiply by, e.g., 3/3, because 3/3 = 1 and so does not change the number. For example, 7/3 x 3/3 = 21/9, whereas 7/3 x 3 = 7, which is three times greater than 7/3!!!
2. And if we have to solve an equation like (x – 4)(x + 5) = 0, we know that either x + 4 must equal 0 or x – 5 must equal 0, so either x must equal -4 or it must equal 5. (Because a number plus its opposite is zero (-4 + 4 = 0) and a number minus itself is zero (5 – 5 = 0).)
3. And if we have to work with a function like f(x) = (x – 4)/(x + 5), then we know x cannot equal -5, because we would then have a zero in the denominator — an impossibility, since we cannot divide anything into zero parts.
4. And if we have to work with an equation like (x – 4)/(x + 5) = 0, then we know the only way to make the equation equal to zero is by making the numerator equal to zero. So (x – 4)/(x + 5) is zero only when x – 4 is zero, which means that x = 4.
The fundamentals of math come up in algebra, precal — and calculus — just as the principle of individual rights comes up in having friends over to watch a movie, in making a movie, and in signing a billion dollar business contract.
Fundamentals and first principles make things more intelligible and easier.
My students learn to take reasoning seriously and to see the importance of reasoning — it makes a world of difference in the student’s life…like the darkness and stagnation of ancient Egypt versus the enlightenment and vitality of ancient Greece.