Read Online Letters to a Young Mathematician by Ian Stewart - Free Book Online
Authors: Ian Stewart
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messy, with lots of loose ends, and the teacher’s job is to impose order on the confusion, to convert a chaotic set of episodes into a coherent narrative. So your learning will be divided into specific modules, or courses, and each course will have a carefully specified syllabus and a text. In some settings, such as some American public schools, the syllabus will specify exactly which pages of the text, and which problems, are to be tackled on a given day. In other countries and at more advanced levels, the lecturer has more of a free hand to pick his or her own path through the material, and the lecture notes take the place of a textbook.
Because the lectures progress through set topics, one step at a time, it is easy for students to think that this is how to learn the material. It is not a bad idea to work systematically through the book, but there are other tactics you can use if you get stuck.
Many students believe that if you get stuck, you should stop. Go back, read the offending passage again;repeat until light dawns—either in your mind or outside the library window.
This is almost always fatal. I always tell my students that the first thing to do is read on. Remember that you encountered a difficulty, don’t try to pretend that all is sweetness and light, but continue. Often the next sentence, or the next paragraph, will resolve your problem.
Here is an example, from my book The Foundations of Mathematics , written with David Tall. On page 16, introducing the topic of real numbers, we remark that “the Greeks discovered that there exist lines whose lengths, in theory, cannot be measured exactly by a rational number.”
One might easily grind to a halt here—what does “measured by” mean? It hasn’t been defined yet, and— oh help—it’s not in the index. And how did the Greeks discover this fact anyway? Am I supposed to know it from a previous course? From this course? Did I miss a lecture? The previous pages of the book offer no assistance, however many times you reread them. You could spend hours getting nowhere.
So don’t. Read on. The next few sentences explain how Pythagoras’s theorem leads to a line whose length is the square root of two, and state that there is no rational number m / n such that ( m / n ) 2 = 2. This is then proved, cleverly using the fact that every whole number can be expressed as a product of primes in only one way. The result is summarized as “no rational number can havesquare 2, and hence that the hypotenuse of the given triangle does not have rational length.”
By now, everything has probably slotted into place. “Measured by” presumably means “has a length equal to.” The Greek reasoning alluded to in such an offhand manner is no doubt the argument using Pythagoras’s theorem; it helps to know that Pythagoras was Greek. And you should be able to spot that “the square root of two is not rational” is equivalent to “no rational number can have square 2.”
Mystery solved.
If you are still stuck after plowing valiantly ahead in search of enlightenment, now is the time to go to your class tutor or the lecturer and ask for assistance. By trying to sort the problem out for yourself, you will have set your mind in action, and thus you are much more likely to understand the teacher’s explanation. It’s much like Poincaré’s “incubation” stage of research. Which, with fair weather and a following wind, leads to illumination.
There is another possibility, but it’s one where help from the teacher is probably essential. Even so, you can try to prepare the ground. Whenever you get stuck on a piece of mathematics, it usually happens because you do not properly understand some other piece of mathematics, which is being used without explicit mention on the assumption that you can handle it easily. Remember the upside-down pyramid of mathematical knowledge? You may have forgotten what a rational number is, or what Pythagoras proved, or how square roots
Because the lectures progress through set topics, one step at a time, it is easy for students to think that this is how to learn the material. It is not a bad idea to work systematically through the book, but there are other tactics you can use if you get stuck.
Many students believe that if you get stuck, you should stop. Go back, read the offending passage again;repeat until light dawns—either in your mind or outside the library window.
This is almost always fatal. I always tell my students that the first thing to do is read on. Remember that you encountered a difficulty, don’t try to pretend that all is sweetness and light, but continue. Often the next sentence, or the next paragraph, will resolve your problem.
Here is an example, from my book The Foundations of Mathematics , written with David Tall. On page 16, introducing the topic of real numbers, we remark that “the Greeks discovered that there exist lines whose lengths, in theory, cannot be measured exactly by a rational number.”
One might easily grind to a halt here—what does “measured by” mean? It hasn’t been defined yet, and— oh help—it’s not in the index. And how did the Greeks discover this fact anyway? Am I supposed to know it from a previous course? From this course? Did I miss a lecture? The previous pages of the book offer no assistance, however many times you reread them. You could spend hours getting nowhere.
So don’t. Read on. The next few sentences explain how Pythagoras’s theorem leads to a line whose length is the square root of two, and state that there is no rational number m / n such that ( m / n ) 2 = 2. This is then proved, cleverly using the fact that every whole number can be expressed as a product of primes in only one way. The result is summarized as “no rational number can havesquare 2, and hence that the hypotenuse of the given triangle does not have rational length.”
By now, everything has probably slotted into place. “Measured by” presumably means “has a length equal to.” The Greek reasoning alluded to in such an offhand manner is no doubt the argument using Pythagoras’s theorem; it helps to know that Pythagoras was Greek. And you should be able to spot that “the square root of two is not rational” is equivalent to “no rational number can have square 2.”
Mystery solved.
If you are still stuck after plowing valiantly ahead in search of enlightenment, now is the time to go to your class tutor or the lecturer and ask for assistance. By trying to sort the problem out for yourself, you will have set your mind in action, and thus you are much more likely to understand the teacher’s explanation. It’s much like Poincaré’s “incubation” stage of research. Which, with fair weather and a following wind, leads to illumination.
There is another possibility, but it’s one where help from the teacher is probably essential. Even so, you can try to prepare the ground. Whenever you get stuck on a piece of mathematics, it usually happens because you do not properly understand some other piece of mathematics, which is being used without explicit mention on the assumption that you can handle it easily. Remember the upside-down pyramid of mathematical knowledge? You may have forgotten what a rational number is, or what Pythagoras proved, or how square roots
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