## "Left As An Exercise To The Reader"

"Left as an exercise to the reader" (LAAETTR) are perhaps the most hated 7 words in all of mathematics. Every student, whether dying to just finish a problem set for a required class or genuinely trying to attain a robust understanding of a new concept, has encountered these infamous words. They represent the worst education has to offer - the tantalizing promise of understanding, swiftly obliterated. And worse, this obliteration is nominally done for the sake of the student. Rigor! The process of learning! You need to figure it out! Nonsense.

All students should be able to spend as much time as they like really thinking through tough problems. The major problem with the LAAETTR method is the total lack of recourse when they simply can't figure them out. Sometimes all it takes is a hint. Other times the operation may be a total loss. Yet without any feedback whatsoever, how is someone who by definition does not know the material supposed to learn anything?

One argument says that the result is already given, so therefore students have a definite endpoint and should be able to figure it out given what they already know. But this is folly. The entire point is for the student to reach the conclusion by a sound proof or order of operations, and thus learn how to reason, not prove some result that is obviously true that the rest of the world is waiting to hear back on despite it having been published in a book already. This approach has no way of correcting for incorrect steps of reasoning, be they algebraic manipulations or successive arguments in a proof. In fact, a student could put forth any chain of reasoning, which so long as it ends up with the desired result one way or the other, may seem convincing.

Another argument is that giving everything to the student will remove any mental challenge, and that it is the intense process of struggling that allows someone to truly understand it. A book may explain how A leads to B, but the student should then figure out how C follows from there. This argument contains a grain of truth - learning lessons the hard way makes them much harder to forget. However, this argument is also nevertheless wrong because it suffers from the same problem as the previous argument - without a way of knowing whether their reasoning is truly correct, how can students determine whether they have, in fact, properly deduced C? The correctness of their reasoning is completely independent of the mental effort it required. This argument also assumes that the struggle will end in a reasonable amount of time, if ever. The issue is economic - after some point it will no longer be worth it to struggle over some problem, as it will inhibit the student from learning anything else, given the finite amount of time available, and especially when further material relies critically on the presently unexplained material. Yet with the exercised fully solved, students can look at the solutions only as much as they need to.

A third argument claims that most students are in a school of some sort that is providing in-person help in the form of professors, TAs, teachers, tutors, or whatever, so the book need not explain everything. But then why write a book? What purpose does the book serve if not to explain? In fact, this argument just shifts the burden of effort from one domain to another - the instructor who omits all of the reasoning when presenting information is guilty of the same exact sin as the LAAETTR book. In fact, this "get someone else to do it" argument implicitly concedes the main point. Namely, it doesn't argue that this kind of in-depth guidance that the book is lacking is not important for learning, but rather that it's just someone else's job to provide it. This is like saying that cars need not come with their own tires - buses come with tires, and commuters should just go take the bus instead. Then why build cars? To wit, what happens when the bus is broken? What happens when there is no bus at all?

Yet another argument, a fourth, is quite cynical - students won't learn anything if they can take the "easy" route. They'll just memorize things or learn only enough to scrape by, and their understanding will be shallower than if they had to exert more effort. First, this argument does not apply to anyone outside of a class. Second, a class that only tests for trivial understanding will only induce students to study for trivial understanding, and this is hardly the book's fault. But even if it tests for meaningful understanding, isn't making this understanding easier to acquire the entire purpose of the class? If a method succeeds at doing that where others don't, what reason is there to persist using less effective methods? Life presents many inescapable opportunities for character-building; education need not impose more than it needs to. Besides, students who have their own reasons for not wanting to learn will not be motivated by a book that has its own reasons for not wanting to teach them. In fact, they're not learning anything with the LAAETTR method anyway, so it looks like the pot is calling the kettle black, isn't it? Perhaps if they had the opportunity to improve themselves, they might take it.

A reasonable justification for the LAAETTR method might be the real-world constraints on book size. The girth of already girthy textbooks may expand to exceed the capacity of any human's strength or wallet to absorb further girth. Yet even this argument falls short as well. Many resources, such as Schaum's Outlines, provide all the practice problems done out! And they are hardly the girthiest books around. It's also the current year, and in the current year we human beans have computers and the internet. A gargantuan corpus of online practice problems adds no extra weight to a person's backpack or bookshelf. In fact, insofar as this argument was ever valid for physical books, it has not been valid in general for many years.

A final argument focuses on the constrained resources of the authors and editors - they can't possible do out all ten zillion practice problems! They have other things they need to do, after all. Once again, this is silly - in order to first craft the practice problems, much less put answers in the back, they'd have to figure out all the steps anyway! But perhaps the cost of type-setting sufficiently advanced material is expensive because it cannot be outsourced so easily. And the proofs for all those practice problems may have long since ceased to exist in any exploitable format. It is here we find the only shred of an argument for LAAETTR - the writer's time is too valuable, unlike your own, so best of luck. It is a far, far cry from any of the arguments presented earlier here in this post that are the ones actually advanced in the LAAETTR method's favor. Luckily, this is what economists call a market opportunity.