Reader’s guide to the Articulated Course · episode 4 / 8 · 3 min 25 s
Learning is learning to solve
Knowing a theorem is not knowing how to solve a problem.
In this episode
Knowing a theorem and knowing how to solve a problem are not the same thing. Fourth episode of the reader’s guide: the method studies. A lesson, a problem, steps from a to e, hints, and a summary you write yourself; the order of a method (factoring) and its traps; the end-of-chapter summary; the methods of volumes II and III.
In the book: “To the reader” (volume I), forewords of volumes II and III; chapter 1 of volume I, method studies 1 to 6 and summary (free extract).
Read the transcript
The first move
Knowing a theorem and knowing how to solve a problem are not the same thing. Faced with a calculation that mixes fractions, powers and parentheses, what is the first move? Learning mathematics is learning to solve problems. And to solve, you need methods: moves that you name, put in order, and repeat. That is the role of the method studies, at the end of each chapter. Articulated Course in Mathematics. Reader’s guide, episode four: learning is learning to solve.
Anatomy of a study
Let’s open a study. It has a number, a level marker, and the name of its method. Here: decoding a calculation that crosses fractions, powers and precedence. First a lesson: the method in a few lines, with the most frequent trap. Then a concrete problem. Then steps, from a to e, that lead the reader one at a time. Hints, when a step is delicate. And the last step calculates nothing: it asks you to sum up the method.
A method is an order
A method is, first of all, an order. Take factoring. One: look for a factor common to all the terms. Two: recognise a standard identity. Three: failing that, group the terms. And the study names the most frequent error: skipping the first step and going straight for an identity. A method is also the list of its traps.
The summary
At the end of the chapter, a summary gathers the methods: one page, one line per method. Precedence: from the deepest level to the most superficial. Common denominator: the least common multiple. Proportionality: a constant coefficient, or a constant product. Factoring: the common factor first. It is the sheet you reread the night before a test, or before tackling a problem that resists.
Methods climb with the volumes
Methods climb with the volumes. In Foundations: the quadratic equation over the complex numbers, basis and dimension of a subspace, the rank theorem, Newton’s method. In Graduate Mathematics: computing a spectrum, the weak formulation, Galerkin’s method, computing curvature. The move stays the same: name the technique, break it into steps, spot its trap.
To remember
To remember. Solving is something you learn. A study is a lesson, a problem, steps, and a summary that you write yourself. And the summary at the end of the chapter holds the methods on one page. The six studies of chapter one are in the free extract, with their summary. The link is in the description. Next episode: the expeditions.
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