Textbooks with solutions and catering to different circumstances

Question

Questions: Are we really taking students into account FULLY when writing textbooks for various areas? Also, are we being unintentionally elitist or dismissive when neglecting to take a more humble approach in writing explanations/proofs and making decisions about providing solutions to exercises?

Let me explain: (using geometry as an example I'm familiar with)

Many textbooks (for (a well known) example, the Topological, Smooth and Riemannian Manifolds texts by Jack Lee) are written with exercises and no solutions to better encourage learning and reduce the undesired yielding to temptation by even more conscientious students etc etc.

My issue with this approach in Math education is that it really stands in the way of self-learners. There is also a tendency to write some books with super concise notation (a common one being little more than a passing mention of 'oh these are Christoffel symbols') in the name of brevity and with the far too common reference to 'mathematical maturity'. My personal feeling about this last bit is that it often gives an escape for simply bad pedagogy. But putting aside personal feelings, I would again say this is a great hindrance to self learners!

Consider how students often progress to become knowledgeable and move on to serious graduate work for instance. A lot of it depends on the school itself. Take a large school like Waterloo or UChicago that offers every course under the sun and the programs are geared towards a more thorough treatment of prerequisites leading to tougher courses (like Symplectic geometry, say). A student in these elite places has access to Profs and fellow students and even seminars that are instrumental to their progress but are not available at smaller schools or certain entire countries. As a result, they have two choices if they hope to try for certain more demanding areas of Math: Move or Chalk it up to impossible.

Isn't it a duty to write in ways that don't (even unintentionally) assume you're talking to either an extraordinary student or one that is smart and has access to the right instructors?

Now I know we can't write amazing textbooks that are self-contained to cater to every student in every circumstance but I believe some obvious steps like opting for full solutions and fleshing out more details even in some 'higher level' texts will have somewhat of a democratizing effect on math education.

Answer 1

Isn't it a duty to write in ways that don't (even unintentionally) assume you're talking to either an extraordinary student or one that is smart and has access to the right instructors?

No. It isn't a duty to write textbooks in a particular way, or to write textbooks at all.

The most common way that advanced textbooks get written is that people teach a course for a while and use that as a basis for a textbook. Unsurprisingly, the primary audience of such a textbook is their own students, and more generally, students who are similarly situated.

It's already the case that there's not a lot of incentive to write textbooks. If it were an expectation that, as part of writing a textbook, one should put in an additional substantial amount of work (like writing full solutions) to make the book less useful to the primary audience, the result would mostly be that people would write fewer textbooks.

It's certainly a good thing when people write for a broader audience, but if you want that to happen, the place to look is at the systemic incentives, not trying to put more obligations on individual people who are already trying to make their subject more accessible.

Answer 2

I believe it is a commercial choice. Universities and public schools prefer to specify such books because it gives the teacher a gatekeeper role in the learning process. In addition, some teachers like to assign graded homework (rather than using tests for all evaluation) and just use the book. A book marketed to self studiers would not keep the answers (not worked solutions, per se) secret. For example my father's WW2 math textbooks ("War Department Education Manuals") all had answers included.

Also, personally, I am a rather undisciplined creature. But the times I have self-studied courses, I had zero problem treating the problems as drill, doing the drill, and using the answers for feedback. Felt no temptation to peek--are you tempted to cheat on reps during a workout on your own? Feedback is important to find mistakes (even silly ones) and also for the psychological "gamification" aspect of seeing how you did. Would play games like "try to get 100%" in my head. In the rare cases that I couldn't solve a drill problem, a glance at the solution is actually kind of a useful hint to then go and try to work the problem. [And of course, since it was drill, I made sure to work every problem, to re-work any missed ones, even if "silly mistake", and to work, without consulting text (while reworking) any problems that I had to [very rarely] sneak a peak at the answer or the text lesson.] Realize that self-study is rather monastic and difficult to stay committed to. So some aspect of feedback is helpful psychologically (and this is an important aspect, do not underestimate it). Furthermore, consider the whole concept of programmed learning, which involves interaction/feedback in a very step by step manner.

I would also add that extremely advanced textbooks (e.g. topology) tend to have small markets and are written as labors of love by people who are rather smart at math (but not at teaching). Often the writers are more interested in explaining to themselves and in not being caught in a mistake of math (similar to Wiki math articles) than they are in the hard slog of actual communication, actual training of the unwashed. And there is not a sufficient market to push the pedants for stronger pedagogy (in contrast to more common/easy subjects).