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Why do most introductory math textbooks not contain solutions? How can a self-learner without access to a university or professor check their work? The common

Why do most introductory math textbooks not contain solutions? How can a self-learner without access to a university or professor check their work? The common answer I get is solutions "rob" the student of learning. Why should the fear of the lazy student cheat those that actually attempt their work and want to see if their solution is correct? Additionally, I heard the argument that "you should know" if your solutions are correct. A beginner is probably prone to subtle mistakes and may think their reasoning is rational. Thus they are probably unaware of any subtle logic errors and could easily fool themselves into thinking their proofs are correct. Is it partially laziness from the authors in creating new problems for homeworks/exams? Or do people think a literal beginner in math can be their own proof checker? I wouldn't want a beginner programmer writing "production ready" code without code review, so why should we expect a beginner math student to write error free proofs? What is the point in not allowing solutions? As great as these text may be, I couldn't and wouldn't recommend them for the beginner. These text seem to be designed solely for a classroom where students will get assistance and feedback from TAs and professors. They seem useless for a beginner to learn on their own without outside proof checking.

chalmette 8y ago
> The common answer I get is solutions "rob" the student of learning. If you study math for some time, you'll realize that people in math community (especially students) like to repeat math related cliches (not necessarily true) over and over again. This is that sort of folk lore. If I were to guess the whys of not including the solutions, I'd say it is to save space as detailed answers to even half the exercises take up considerable space; also "laziness" because writing detailed, explanatory answers is no different from writing the rest of the book. Who wants to do double the work ? :
ghc35365 8y ago
Just wanted to add MIT notes on discrete mathematics have the solution manual.
jimhefferon 8y ago
> Why do most introductory math textbooks not contain solutions? I offer a couple of Free texts, one of which is pretty popular (http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/). The answers, completely step-by-step solved, are available. I can give you two reasons. 1) Preparing good questions and giving answers is about half the work. On my page you'll see that the book is 500 pages and the answers are 400 pages. And it is not the half that is fun, or that gets you credit. I find that compiling the answers greatly increases the quality of the book because I keep going back and adjusting the presentation, etc. But others don't think that. Rather the opposite; I have been told that it is work only appropriate for grad students. 2) I have gotten a fair number of emails along the lines of "I'd like to adopt your book but since anyone can download the answers, I cannot teach out of it." It has not in the past proven to be a fruitful strategy for me to inform the email writer that the first thing any 2018 undergrad does on getting a text asignment is to google for the answers pdf, and that those students always succeed. So one reason to not provide answers is to get more adoptions. This ties in with (1) because my five year reviews are not impressed that random self-learners find the text useful. They ask about adoptions.
- dvfjsdhgfv 8y ago
  I have admit it's one of the nicest math books laid out in LaTeX, it's a pleasure read.
  - jimhefferon 8y ago
    That's kind of you; thank you.
- sondr3 8y ago
  As the other commenter mentioned, it's beautifully typeset, do you have any hints for someone who writes a fair bit of math in LaTeX?
  - jimhefferon 8y ago
    I think of it as very generic LaTeX. I use the AMS math styles for formatting the equations. The fonts are not the standard ones, but they were chosen for xerox-ability rather than some aesthetic (which I don't have). Other than that it is straightforward. (The LaTeX is at https://gitlab.com/jim.hefferon/linear-algebra https://gitlab.com/jim.hefferon/linear-algebra .)
semigroupoid 8y ago
One reason might be that exercises that ask for proofs often have several correct answers. It would probably be just as confusing to a beginning student to come up with a proof for some exercise and then see the author take a completely different approach in the provided solution. The student still wouldn't know whether his proof is correct or not.
throwawaymath 8y ago
I hear you on how frustrating that is. There are several reasons they don't contain solutions. The reasons you have heard are idealistic but practically incorrect. In particular this is wrong, for the reasons you intuited: > Additionally, I heard the argument that "you should know" if your solutions are correct. Anyone who believes this has forgotten their own early struggles or never watched beginning students struggle with something comparatively basic, like induction. In fact, introductory (proof-based) math courses are particularly dangerous for not knowing whether or not your proof is sound, because you've not yet built the mathematical maturity to check your own work. In a similar vein, beginning students often have a high level mental model of why a statement is true and which "tools" are needed, but no idea how to concretely put these together in a sound proof. This is more of an opinion that doesn't reflect reality: > The common answer I get is solutions "rob" the student of learning. If the presence of the solution means the student will simply refer to it instead of e.g. completing homework on their own, then the student was probably never really engaged with the material in the first place. It's one thing to laboriously complete the proof first and then use the solution to check it, but it's another thing entirely if the student is just skipping all the work. This is especially true nowadays, because students who want to get easy answers can just go on math.stackexchange for virtually any homework assignment. These answers are parroted because there is a deep seated "trial by fire" culture in advanced math. But this doesn't have anything to do with why the solutions don't tend to be in the books. To begin with, you have to check your assumptions: > How can a self-learner without access to a university or professor check their work? These textbooks are (for better or worse) not designed to be used by self-learners. In fact there isn't really any mainstream publication effort in higher mathematics targeting autodidacts. Therefore that criticism - while valid - isn't really applicable to math textbooks. Ostensibly students are using these textbooks under the instruction of a professional mathematician in a classroom setting. The other reason mostly follows from the first one. The proof of any given statement is typically significantly longer than the statement itself. As it stands, math textbooks are usually filled with sequences of axioms, definitions, propositions, theorems and lemmas. Each chapter tends to have at least six or so of those sequences, but potentially dozens. The proofs of theorems are included, which means that all but the most trivial of chapter exercises would significantly lengthen the entire enterprise. That's a lot more writing, grammatical editing, mathematical proof-reading, page binding, etc for the whole publication process.
mlevental 8y ago
are you complaining about this book or books like this in general? this book has solutions to all "assignments" and sample exams...?
hinkley 8y ago
Because the books aren't for self-taught people. They're to sell a crapload of books indirectly through University professors. As for the "you should know" angle, I do believe that a lot of problems chosen for math classes are chosen because they are asymmetric. That is, getting the answer is much harder than checking the answer. For fractions or systems of equations you absolutely can check your work. It's frustrating for me that nobody appears to be teaching kids that. If you want to get an A on a college level math test your best use of time left over at the end of the test is to check your work.
coliveira 8y ago
Because books with solutions give a false sense of security. The goal of exercises is to have the students to think for themselves about the solution. It really doesn't matter if they can solve the problem perfectly. People concerned about solutions are in fact thinking about a quick way to get the best grades for an exam. Which is fine, but it should not the first goal of anyone writing a text book on a mathematical topic.
globuous 8y ago
You're completely right. Although ome have solutions book. For math specifically, they all state theorems and prove them. It's good practice to try to prove a stated theorems without looking at its proof first.
smadge 8y ago
Most math textbooks are full of proofs of theorems and examples solving problems. This is not better than direct feedback from an expert on your own proofs (e.g. a tutor, professor, or TA), but it is in my opinion exactly what you are talking about. Students should read the proofs and the examples, and try it themselves until they really understand it. Then the exercises usually fall into place.
Francute 8y ago
> How can a self-learner without access to a university or professor check their work? Their "work" means: Find a solution, AND then check it. So, not checking it by yourself is being lazy too. Also, if you are learning; software like wolframalpha can help you too to check your solution. [Thanks, throwawaymath for recall this basic answer]. > I wouldn't want a beginner programmer writing "production ready" code without code review, so why should we expect a beginner math student to write error free proofs? "Code Reviews" are for checking "how did the programmer solved that problem". On the other side; you know if some code is giving good or bad outputs by checking it with "Tests" (In the worst scenario; manually). So, you have to learn to "test" your solution. The creativity is in how you managed to get the solution and teachers should give insights about your way to solve it and present you easier or better ways to achieve the same. (Just change "teachers" for "senior devs" and now, that is code review). > These text seem to be designed solely for a classroom where students will get assistance and feedback from TAs and professors. Probably that's true. You have to consider those types of books as "slides"; just like a guide for the teacher and memory refresh for students.
eastWestMath 8y ago
Well yeah, you need to be pretty far along in your mathematical education before you can learn proof based mathematics on your own. One thing I notice when marking undergraduate work is I can tell they mean the right thing, but what they actually wrote is absolutely wrong. A self learner, even armed with a solution manual, will not fix those mistakes.
hackermailman 8y ago
Author of this other math book explains why he didn't include solutions, to promote inquiry based learning, with links to research papers from a grad class on evidence based teaching https://infinitedescent.xyz/20180214-exercises-ibl/ https://infinitedescent.xyz/20180214-exercises-ibl/ "people learn more when they discover something for themselves than they do if someone tells them about it." The whole idea of IBL is apparently just even communicating properly what problem you're having is a better learning experience than looking at an answer. As an anecdote a few times when I've asked math.stackexchange I found the answer just by thinking about my question.
- deleted 8y ago
  [deleted]

chalmette 8y ago

> The common answer I get is solutions "rob" the student of learning. If you study math for some time, you'll realize that people in math community (especially students) like to repeat math related cliches (not necessarily true) over and over again. This is that sort of folk lore. If I were to guess the whys of not including the solutions, I'd say it is to save space as detailed answers to even half the exercises take up considerable space; also "laziness" because writing detailed, explanatory answers is no different from writing the rest of the book. Who wants to do double the work ? :

ghc35365 8y ago

Just wanted to add MIT notes on discrete mathematics have the solution manual.

jimhefferon 8y ago

> Why do most introductory math textbooks not contain solutions? I offer a couple of Free texts, one of which is pretty popular (http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/). The answers, completely step-by-step solved, are available. I can give you two reasons. 1) Preparing good questions and giving answers is about half the work. On my page you'll see that the book is 500 pages and the answers are 400 pages. And it is not the half that is fun, or that gets you credit. I find that compiling the answers greatly increases the quality of the book because I keep going back and adjusting the presentation, etc. But others don't think that. Rather the opposite; I have been told that it is work only appropriate for grad students. 2) I have gotten a fair number of emails along the lines of "I'd like to adopt your book but since anyone can download the answers, I cannot teach out of it." It has not in the past proven to be a fruitful strategy for me to inform the email writer that the first thing any 2018 undergrad does on getting a text asignment is to google for the answers pdf, and that those students always succeed. So one reason to not provide answers is to get more adoptions. This ties in with (1) because my five year reviews are not impressed that random self-learners find the text useful. They ask about adoptions.

dvfjsdhgfv 8y ago
I have admit it's one of the nicest math books laid out in LaTeX, it's a pleasure read.
- jimhefferon 8y ago
  That's kind of you; thank you.
sondr3 8y ago
As the other commenter mentioned, it's beautifully typeset, do you have any hints for someone who writes a fair bit of math in LaTeX?
- jimhefferon 8y ago
  I think of it as very generic LaTeX. I use the AMS math styles for formatting the equations. The fonts are not the standard ones, but they were chosen for xerox-ability rather than some aesthetic (which I don't have). Other than that it is straightforward. (The LaTeX is at https://gitlab.com/jim.hefferon/linear-algebra https://gitlab.com/jim.hefferon/linear-algebra .)

semigroupoid 8y ago

One reason might be that exercises that ask for proofs often have several correct answers. It would probably be just as confusing to a beginning student to come up with a proof for some exercise and then see the author take a completely different approach in the provided solution. The student still wouldn't know whether his proof is correct or not.

throwawaymath 8y ago

I hear you on how frustrating that is. There are several reasons they don't contain solutions. The reasons you have heard are idealistic but practically incorrect. In particular this is wrong, for the reasons you intuited: > Additionally, I heard the argument that "you should know" if your solutions are correct. Anyone who believes this has forgotten their own early struggles or never watched beginning students struggle with something comparatively basic, like induction. In fact, introductory (proof-based) math courses are particularly dangerous for not knowing whether or not your proof is sound, because you've not yet built the mathematical maturity to check your own work. In a similar vein, beginning students often have a high level mental model of why a statement is true and which "tools" are needed, but no idea how to concretely put these together in a sound proof. This is more of an opinion that doesn't reflect reality: > The common answer I get is solutions "rob" the student of learning. If the presence of the solution means the student will simply refer to it instead of e.g. completing homework on their own, then the student was probably never really engaged with the material in the first place. It's one thing to laboriously complete the proof first and then use the solution to check it, but it's another thing entirely if the student is just skipping all the work. This is especially true nowadays, because students who want to get easy answers can just go on math.stackexchange for virtually any homework assignment. These answers are parroted because there is a deep seated "trial by fire" culture in advanced math. But this doesn't have anything to do with why the solutions don't tend to be in the books. To begin with, you have to check your assumptions: > How can a self-learner without access to a university or professor check their work? These textbooks are (for better or worse) not designed to be used by self-learners. In fact there isn't really any mainstream publication effort in higher mathematics targeting autodidacts. Therefore that criticism - while valid - isn't really applicable to math textbooks. Ostensibly students are using these textbooks under the instruction of a professional mathematician in a classroom setting. The other reason mostly follows from the first one. The proof of any given statement is typically significantly longer than the statement itself. As it stands, math textbooks are usually filled with sequences of axioms, definitions, propositions, theorems and lemmas. Each chapter tends to have at least six or so of those sequences, but potentially dozens. The proofs of theorems are included, which means that all but the most trivial of chapter exercises would significantly lengthen the entire enterprise. That's a lot more writing, grammatical editing, mathematical proof-reading, page binding, etc for the whole publication process.

mlevental 8y ago

are you complaining about this book or books like this in general? this book has solutions to all "assignments" and sample exams...?

hinkley 8y ago

Because the books aren't for self-taught people. They're to sell a crapload of books indirectly through University professors. As for the "you should know" angle, I do believe that a lot of problems chosen for math classes are chosen because they are asymmetric. That is, getting the answer is much harder than checking the answer. For fractions or systems of equations you absolutely can check your work. It's frustrating for me that nobody appears to be teaching kids that. If you want to get an A on a college level math test your best use of time left over at the end of the test is to check your work.

coliveira 8y ago

Because books with solutions give a false sense of security. The goal of exercises is to have the students to think for themselves about the solution. It really doesn't matter if they can solve the problem perfectly. People concerned about solutions are in fact thinking about a quick way to get the best grades for an exam. Which is fine, but it should not the first goal of anyone writing a text book on a mathematical topic.

globuous 8y ago

You're completely right. Although ome have solutions book. For math specifically, they all state theorems and prove them. It's good practice to try to prove a stated theorems without looking at its proof first.

smadge 8y ago

Most math textbooks are full of proofs of theorems and examples solving problems. This is not better than direct feedback from an expert on your own proofs (e.g. a tutor, professor, or TA), but it is in my opinion exactly what you are talking about. Students should read the proofs and the examples, and try it themselves until they really understand it. Then the exercises usually fall into place.

Francute 8y ago

> How can a self-learner without access to a university or professor check their work? Their "work" means: Find a solution, AND then check it. So, not checking it by yourself is being lazy too. Also, if you are learning; software like wolframalpha can help you too to check your solution. [Thanks, throwawaymath for recall this basic answer]. > I wouldn't want a beginner programmer writing "production ready" code without code review, so why should we expect a beginner math student to write error free proofs? "Code Reviews" are for checking "how did the programmer solved that problem". On the other side; you know if some code is giving good or bad outputs by checking it with "Tests" (In the worst scenario; manually). So, you have to learn to "test" your solution. The creativity is in how you managed to get the solution and teachers should give insights about your way to solve it and present you easier or better ways to achieve the same. (Just change "teachers" for "senior devs" and now, that is code review). > These text seem to be designed solely for a classroom where students will get assistance and feedback from TAs and professors. Probably that's true. You have to consider those types of books as "slides"; just like a guide for the teacher and memory refresh for students.

eastWestMath 8y ago

Well yeah, you need to be pretty far along in your mathematical education before you can learn proof based mathematics on your own. One thing I notice when marking undergraduate work is I can tell they mean the right thing, but what they actually wrote is absolutely wrong. A self learner, even armed with a solution manual, will not fix those mistakes.

hackermailman 8y ago

Author of this other math book explains why he didn't include solutions, to promote inquiry based learning, with links to research papers from a grad class on evidence based teaching https://infinitedescent.xyz/20180214-exercises-ibl/ https://infinitedescent.xyz/20180214-exercises-ibl/ "people learn more when they discover something for themselves than they do if someone tells them about it." The whole idea of IBL is apparently just even communicating properly what problem you're having is a better learning experience than looking at an answer. As an anecdote a few times when I've asked math.stackexchange I found the answer just by thinking about my question.

deleted 8y ago
[deleted]