FAQ begins here:
PROBLEMS VERSUS EXERCISES
I frequently encounter discussions among parents about repetitive school math lessons, so a few years ago I prepared this Frequently Asked Question (FAQ) document about the distinction between math exercises (good in sufficient but not excessive amount) and math problems (always good in any amount).
Most books about mathematics have what are called "exercises" in them, questions that prompt a learner to practice the concepts discussed in the mathematics book. By reading one mathematics book, and then several more, I learned that some mathematicians draw a distinction between "exercises" and "problems" (which is the terminology generally used by the mathematicians who draw this distinction). I think this distinction is useful for teachers and learners to consider while selecting materials for studying mathematics, so I'll share the quotations from which I learned this distinction here. I first read about the distinction between exercises and problems in a Taiwan reprint of a book by Howard Eves.
"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy.
"It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."
Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.
I have since read about this distinction in several other books.
"Before going any further, let's digress a minute to discuss different levels of problems that might appear in a book about mathematics:
Level 1. Given an explicit object x and an explicit property P(x), prove that P(x) is true. . . .
Level 2. Given an explicit set X and an explicit property P(x), prove that P(x) is true for FOR ALL x [existing in] X. . . .
Level 3. Given an explicit set X and an explicit property P(x), prove OR DISPROVE that P(x) is true for for all x [existing in] X. . . .
Level 4. Given an explicit set X and an explicit property P(x), find a NECESSARY AND SUFFICIENT CONDITION Q(x) that P(x) is true. . . .
Level 5. Given an explicit set X, find an INTERESTING PROPERTY P(x) of its elements. Now we're in the scary domain of pure research, where students might think that total chaos reigns. This is real mathematics. Authors of textbooks rarely dare to pose level 5 problems."
Graham, Ronald, Knuth, Donald, and Patashnik, Oren (1994). Concrete Mathematics Second Edition. Boston: Addison-Wesley, pages 72-73.
This digression becomes the subject of a, um, problem in Exercise 4 of Chapter 3: "The text describes problems at levels 1 through 5. What is a level 0 problem? (This, by the way, is NOT a level 0 problem.)"
"First, what is a PROBLEM? We distinguish between PROBLEMS and EXERCISES. An exercise is a question that you know how to resolve immediately. Whether you get it right or not depends on how expertly you apply specific techniques, but you don't need to puzzle out what techniques to use. In contrast, a problem demands much thought and resourcefulness before the right approach is found. . . .
"A good problem is mysterious and interesting. It is mysterious, because at first you don't know how to solve it. If it is not interesting, you won't think about it much. If it is interesting, though, you will want to put a lot of time and effort into understanding it."
Zeitz, Paul (1999). The Art and Craft of Problem Solving. New York: Wiley, pages 3 and 4.
". . . . As Paul Halmos said, 'Problems are the heart of mathematics,' so we should 'emphasize them more and more in the classroom, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are.'
"The problems we have selected are definitely not exercises. Our definition of an exercise is that you look at it and know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises."
Andreescu, Titu & Gelca, Razvan (2000), Mathematical Olympiad Challenges. Boston: Birkhäuser, page xiii.
"It is easier to advance in one topic by going ahead with the more elementary parts of another topic, where the first one is applied. The brain much prefers to work that way, rather than to concentrate on ugly technical formulas which are obviously unrelated to anything except artificial drilling. Of course, some rote drilling is necessary. The problem is how to strike a balance."
Lang, Serge (1988), Basic Mathematics. New York: Springer-Verlag, p. xi.
"Learn by Solving Problems
"We believe that the best way to learn mathematics is by solving problems. Lots and lots of problems. In fact, we believe the best way to learn mathematics is to try to solve problems that you don't know how to do. When you discover something on your own, you'll understand it much better than if someone just tells it to you.
. . . .
"If you find the problems are too easy, this means you should try harder problems. Nobody learns very much by solving problems that are too easy for them."
Rusczyk, Richard (2007). Introduction to Algebra. Alpine, CA: AoPS Incorporated, p. iii.
First, I don't know any mathematician personally who makes such a clear linguistic distinction between 'exercise' and 'problem'. Once you get to university-level mathematics, many exercises are problems in your sense but they still tend to be called exercises or something similar. If you insist on this terminological divide, I doubt most people will understand you.
Secondly, there is the matter of an exercise's pedagogical purpose. Is it to sharpen general problem solving skills or to enlighten the student on a conceptual level? This goes beyond difficulty. It's a false dichotomy when stated so simply, but there is still something there. Many IMO-style problems are conceptually barren but still very tricky to solve. Conversely, some of my most enlightening learning experiences were solving guided sequences of exercises in a mathematical form of Socratic learning where none of the steps were individually too hard but still involving enough that they forced me to think and thus develop some insight on my own. (This approach can also fail. Silverman's otherwise excellent book Rational Points on Elliptic Curves has a guided proof of Bezout's theorem in the appendix that is just too atomized to engender much understanding.)
The "mathematical form of Socratic learning where none of the steps were individually too hard but still involving enough that they forced me to think and thus develop some insight on my own" is what I attempt to provide in my live, face-to-face mathematics classes. I'm not worried about Khan Academy reducing the market for those classes (and in fact encourage current and prospective students to try out Khan Academy) because providing that sort of instruction is very hard to automate. As your example of Silverman's book points out, it is more of an art than a settled science to decide just how many steps to show with Socratic guidance, not to mention that different learners need different steps drawn out for them.