# What other mathematicians have advocated for freely publicizing detailed solutions, to every exercise?

Besides the 2 mathematicians quoted below and me, who else has touted free dissemination to students of detailed solutions, to EVERY exercise and problem (like in textbooks)? I uphold this wholeheartedly! This free dissemination ought to be the norm!

I do NOT refer to snippety one-line answers at the back of a textbook, student solution manuals that solve merely some or half of the exercises, or solution manuals restricted to instructors.

1. Robert Ash (1935-2015), Preface to Real Variables with Basic Metric Space Topology.

I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in elementary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the book suitable for independent study, and further widens the audience.

1. David Patrick, Introduction to Counting and Probability (2005), page v.

However, if you are using this book on your own to learn independently, then you probably have a copy of the solution book, in which case there are some very important things to keep in mind:

1. Make sure that you make a serious attempt at the problem before looking at the solution. Don't use the solution book as a crutch to avoid really thinking about a problem first. You should think hard about a problem before deciding to give up and look at the solution.

2. After you solve a problem, it's usually a good idea to read the solution, even if you think you know how to solve the problem. The solution that's in the solution book might show you a quicker or more concise way to solve the problem, or it might have a completely different solution method that you might not have thought of. [emboldening mine]

3. If you have to look at the solution in order to solve a problem, make sure that you make a note of that problem. Come back to it in a week or two to make sure that you are able to solve it on your own, without resorting to the solution.

• Jim Hefferon's Linear Algebra book does this (hefferon.net/linearalgebra/index.html), though I can't find the place where he explicitly endorses it. However, all the verbiage he uses assumes that the student has access to it (which they do, without "cheating"). Nov 5, 2022 at 13:43
• As @kcrisman says, I provide fully worked answers to all exercises. Just to have said it, though, I'll note a couple of things. First, I have had instructors tell me they would use the text except that all the answers are available. They want to grab questions for quizzes or hand-ins from the text. And second, writing those exercises and answers is much more work than writing the text body. For my latest book, writing the answers took a year. Nov 22, 2022 at 16:12

I think if you look at texts that are more designed for self study and marketed to students, it's a norm. Schaum's Outlines does this. Even more extreme, perhaps there are programmed instruction texts. I know my father's WW2 education manuals (math from geometry to calculus) were all like this also. Some individual texts like this are Thomas and Thomas Finney Calculus (at least from when they were alive) or Tannenbaum and Pollard ODE or Wienberger PDE books.

I think the big driver behind keeping the solutions cloaked is the market being teachers, not students, when texts are selected. (Giving graded homework, wanting to force students not to look at answers to fast, or wanting to retain more of a role for being.)

If you happen to be a student or meet one, who really doesn't want to have solutions or answers available (incapable of discipling self to work the problems before looking at the answer), than advise him to tear out the pages at the back and burn them. Personally, I think having the solutions or at least answers is very helpful. So, that's silly to say a text is worse by having solutions. But if you really think so, just burn those pages.

I think solutions or at least answers gives a feedback loop, including for "dumb mistakes". Can even be a form of partial hint (helps to know the answer to work the problem, look at the integral stumpers on MSE). And this is a feedback loop (or hint aid) that is immediately available, very important in the learning process, rather than waiting for hours or days.

• "Personally, I think having the solutions or at least answers is very helpful." I agree! You are dead right! I edited my post to clarify. "So, that's silly to say a text is worse by having solutions. But if you really think so, just burn those pages." Yes, this is silly to say. I do not think this at all! Nov 6, 2022 at 4:50

This is hardly "touting" and surely below the level of your interest, but I will nevertheless mention that two books I wrote at the high-school / early college level both include solutions to every exercise at the back of the book:

• How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra, Cambridge, 2011.
• Pop-Up Geometry: The Mathematics behind Pop-Up Cards, Cambridge, 2022.