Why Math Books Don't Have Answers
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- เผยแพร่เมื่อ 13 ก.ย. 2024
- In this video I talk about why math books don't have answers to all of the exercises. I give 3 reasons why books don't have all of the solutions. Do you have any opinions? If so, please leave a comment below.
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One downside of not having answers is that self learners can convince themselves that they have a correct understanding of a problem while being off base. Even being slightly off with math can have catastrophic consequences as you learn more
Specially when major theorems are given as exercises
Right
Not that catastrophic self studying means you actually make mistakes quite frequently and have to review your knowledge continuously.
Seems as though those who have formal education also have gaps and weaknesses in their knowledge as they progress. May be something that everybody suffers from, so having to go back and shore up things is inevitable. Especially considering that you can't master everything before being able to move on, and you never know if you've covered something to a sufficient extent
👍I experienced it first hand. After Plane Trigonometry volume by S L Loney, I got thirsty for his Analytical trigonometry, however finding several exercises beyond my reach, so skipped over them. Later I found solutions to both volumes available as separate books, probably by different authors. Having a guide accessible when needed was a huge motivation, that I was not alone in a wilderness, it really simulated my self-learning desire for several other Math branches.
Ivan Niven in one of his shortest books says the way to learn Math is by doing it ... And for self-learning, guides are a huge support when directions are needed at some steps☺
My very personal opinion about mathbooks not having all solutions is because publishers want to make some money more by selling a separate solutions book.
I've seen solutions manuals offered "free" to teachers/professors who adopt a textbook for their classes, but I haven't seen them for sale in any way that would make me think they were a significant source of profit for publishers.
That is actually most obvious answer, likely the only correct one.
Yup you got that right
@@VTGDFGGGT55RTT yeah this happens very often 🤣😅🤣
@Kunal Rana 🤪
Another thing to comment: I had a professor who give us homework with the final solutions to every problem. We just have to show our work that leads to that final answer. I liked this approach, because it helps us stay on the right path of solving problems, and gives us an opportunity to "debug" through the work when we get different results than the final solutions. This type of skill was so transferable in helping me become a better at writing code, where debugging is a important process. Back to the point, I think having the just final answers is what keeps us from going astray.
If you put it that way it makes a lot of sense, I didn't relate debugging with maths before and what you say is 100% accurate.
It's not very easy to give the final answer to "Show that the square root function is uniformly continuous" without giving away the whole solution.
Most undergraduate and above math books are about proving theorems. The final answer is already given: the fact that it is true.
@@MasterHigure @139 Proving theorems already gives you that "solution", that is just the nature of a proof problem. In a computational problem, you are asked: "If Q, then what is P?" In a proof problem, you as asked to prove "If Q, then P". In both cases, you are filling in the details, the 'showing your work', the steps in the proof to get to the "solution", P.
If you think about it, computational problems and proof problems are actually the same thing. It's just that when doing proofs, you are 'showing work' for a more *general* case.
The *true solution* is the 'showing your steps' that leads to the final result. That is something where we need to critically think and be creative, since you are doing problem solving.
For an educational viewpoint, the point is to learn the subject. Without *any* solution, there is no feedback. That is fine when grading, because professors want students to try and then give feedback. For self-learning, it's more difficult as I explained in more detail in my other comment.
@@Diaming787 exactly
I had the same question. Answers are exceedingly important for self-learners.
definitely!
I agree
Math grad student here. I can tell you that in my experience, it is immensely helpful to have solutions in the back of the book. They do not need to be elaborate; just knowing whether the final answer I arrived at is correct or not is extremely helpful. In most of my classes, homework was worth only ~10% of the grade and was graded mostly on completion, so even if a student did just copy answers, it wouldn't affect their grade enough to make up for doing poorly on the tests. I would encourage all professors to choose books that have at least some solutions in the book so students have an idea as to whether they're understanding things correctly or not.
It's important to keep in mind that most books are written by professors for use in class. That's why explanations are usually concise and minimal, and solutions are lacking. They're presupposing that the student has a teacher to use as a complimentary resource. That of course makes it difficult to self-study, but I think it's natural for professors to make that assumption, and it's also less work on them since they have less writing to do.
When authors take a different approach and target self-studying students, however, the book's approach changes drastically. Simply look at books like How to Prove it or Calculus for Dummies, or Calculus Made Easy. Yes, they don't all have solutions to everything, but the text makes up for it. It's also why these books become rare as the topic becomes more advanced, since few people outside of academia would read these works.
For self-learners, we would at least like to have selected answers and hints. If we come up with a solution we think is right but there is the subtle nuance we missed, it would be bad in the long run. I see having answers as a way to have a *feedback* needed for mastery. It's one of those tricky balance between giving yourself to struggle enough, but not too much.
Luckily, looking up solutions on internet is not hard to find. Point is, there is a difference between using solutions to cheat yourself, and using answers as a feedback to facilitate mastery.
I think we should treat Math like programming,
Programmer use stackoverflow all the time, not because they're stupid or like to cheat.
It's because solving problem is more important than debugging things one by one until for who knows days to months you found the solution.
Even with compiler and debugger giving you feedback, it's still took a long time to solve a problem from scratch.
And programmers often learn from someone elses solution, and learning from other people solution often was better than coming up with your own solution.
And by learning from someone elses solution, you learn to integrate their solution and improve their solutions
We're all have a limited time and only 5 working days, there are no time to solve a problem exactly like Euler did, solving problem the way he originally did.
That kind of thinking is slowing people down
Most books do have selected answers. And a decent hint is often given in the examples in the text itself.
And treating math like programming where we just end up looking at stack overflow doesn’t lead to learning (just like a programmer that does that all day hasn’t learned anything either - maybe they completed their project and got paid but they didn’t become better). Actually the more I teach the more I come to think that even if we don’t even grade math homework, providing all the solutions up front is hurting students more than helping them.
@@stephenbeck7222 i'm disagree with you.
because although we're often use stackoverflow, we have to dissect the code to understand how its work, it's only when you understand how things work, then you can implement it in your own way.
you can't just copy and paste someone else code from stackoverflow.
you have to understand stuff to be able to implement stuff. even dissecting the already solved solution will leads you to understanding.
so, solving the problem from zero isn't the only way to understand stuff, it's ineffective.
as long as you can implement the concept, you earn the skill
i get better in programming just by doing that.
and let me tell you that i'm not your typical run of the mill programmer that work on CRUD app all day.
i'm working with math heavy graphic programming, machine learning, signal processing, image processing and AI.
Never underestimate stranger on the internet.
here the thing,
AI gets better because we fed them billions of datasets. they improve from working on sheer quantity of data.
yet when it came to human, we are expected to spent 1 days to weeks on only a handful of problems.
because we're not allowed "cheat"
we already knew that AI improved from being fed with sheer quantity of datasets,
yet we, the creator didn't have that privilege of accessing billions of math problem because someone think it's cheating, it's indignity and shameful.
we're sabotaging ourself with our outdated way of thinking.
we need to let go this false belief of "people won't improve if we gave them all of the solution", and instead we should put our faith in them.
it's not our place to stunt their growth by forcing our belief to to them.
most of the time, by giving people access to resources, or open up resources like "open source project", it help people bloom and it let them improve higher than ever before.
Perhaps learning math isn't meant to be a sole endeavour? Perhaps we actually need a mentor or teacher to guide us
@@jensenraylight8011 thanks for putting this into words and i completely agree, ever since i really picked up programming i relearned how to learn things, the textbook way may work good for some people but it feels terribly inefficient and somewhat of an outdated approach to me. just open source anything really
The reason why I like solutions is because they often fill in the gaps for me;
I _have_ in fact developed a genuinely great understanding of a lot of math concepts by simply checking one single step in a solution, and then realising that "hey yeah, that's right, they use that step because that is how it actually works in this particular exercise", and then I can apply this to later problems, and develop that understanding more and more from there. Some people just learn better that way.
For me personally, I enjoy self studying mathematics so of course when I pick out a math book that I am going to work through it is almost always a deal breaker when the book does not have answers. I really like to be able to check my work after I have worked through a problem to see if I did the problem correctly. I feel like if I were to work through a math book that did not contain answers, the uncertainty as to if I worked through the problems correctly or not would eat me alive.
Yeah, this is exactly the way I feel about it
What if your solution was different than the one in the textbook?
@@AnyVideo999 For me the only purpose of the book solutions is to provide verifiability - I do not feel comfortable learning material without any kind of check and balance except my own feeling of internal consistency.
Obviously I don't have to follow the book solution and it is maybe not even the best solution, but seeing it allows me to at least assign some degree of probably true/ probably false to my own result.
And when there are numerical answers it is almost a perfect verification tool
John Jackson wrote the most widely used graduate level Electricity & Magnetism text. Some of it was so difficult that when he died students said: "Now we'll *never* know the solutions to the hardest problems!"
Wouldn’t that suggest that the book was insufficient in its learning content so as to answer the problems correctly?
@@the_abandoned_monastery7218 sure….but that wasn’t really the point lol
@@the_abandoned_monastery7218 this question is answerable when you the read the book itself
@@AlexTalksTwo I see, it was more of an act of praise than anything else. Got it
@@the_abandoned_monastery7218 not really, just knowing all the things required doesn't mean u can solve the hard problems and that ability cannot be delivered by any book
As an instructor, I always preferred to assign a mixture of odd- and even-numbered exercises as homework: some that had answers in the back of the book, so the students could check right away if they were on the right track, and some that didn't, so that I could see what kind of answers the students were coming up with when they didn't have the correct answers to check against or copy from.
You'll find out what kind of answers they come up with during tests. Why give students questions without answers while they are trying to learn new material? If students want to just look at the answers just to get the homework done, it will be a really good lesson to learn to not do that when they start failing tests.
@@tyh7388 given the grade significance tests have at college level (some lecturers I've had at college level give 3 or 4 exams in the entire semester), not allowing students to realize they may be becoming over-reliant on the answer sheet until their first one is a poor grade does not seem optimal to me.
Obviously this could be resolved partially with a fully reformed curriculum setup but I don't know if that is really in the scope of this discussion
@@fatman3762 I can see how answers could hurt students. Couldn’t you rip out the answers from those textbooks for students taking those classes? That should solve the over-reliance issue as well as emphasize the point that students shouldn’t rely on the answers at that level. My argument is that the textbooks will be used by more than students in colleges who don’t have a teacher to rely on. Tutors are too expensive. Having the answers is the next best thing for self learning.
It is great to have solutions (and proofs) for all the exercises, but as a separate book. Also it is a great idea to have hints (lots of hints) at the end of the book
definitely!!
That is exactly what the Art of Problem Solving books do!
More money for the publisher!
So instead of buying what should only be 1 book, you now have to buy 2? 🤦
I agree 100%. If you are working on a problem, and you have a second book with solutions, it would put another action of yours, getting the book and checking the answer.
In my opinion, it would make sense to randomize the order of the solutions in book 2 (solutions )for the problems in book 1 (workbook, standard book). If you have to go through the struggle to find the answer for let's say problem number 674 of 1000 or even more problem sets in the solution book, it would probably be enough of a "struggle" for you to prevent you from searching before doing the math yourself, transforming into the mindset to only search it, after you've tried everything possible to solving let's say question nr. 674, that you would search the solution only after trying it. Who would like to struggle searching a book of let's say more than a thousand pages of only solutions, if the bookpage for the solution of each problem is randomized? Even if you would search just 300 pages, it would probably take you at least 60 minutes of searching for the same keywords if you are able to scan through 5 pages per minute average speed. With such a system in mind, only the desperate enough persons - or those who tried everything they can, would have the willpower to search until they find the solution.
To make this even harder, the answers may not even be numbered, just indicated with dots to differentiate between 2 different problems (problem number x, dot, problem number y, where x and y are randomized natural positive numbers in the range of the number of problems).
Try finding the right solution if you have to look for multiple keywords to align and also perfectly make sense to the question in the first place.
I think that this would be the most fair system for both students who are willing to learn (and not cheat) and other students who cheat and have to go through that struggle mentioned above.
Even if they go through that struggle, they might learn something new while scanning through the answers with their eyes.
And the not-cheating students, who really just want to check the answer might learn something new too, while searching in the book.
This is one reason I really like the Larson calculus books; they have a very helpful and extensive website which not only has the answers, but the details of how they were worked out. And for practically every edition of the books. For a self-study student that's invaluable. I wish there were similar helps for more advanced math books, like Fraleigh's Abstract Algebra!
Larson calculus is a really great book for mass market, and in general the other books from Cengage publishing are good also
Is Fraleigh better than Hungerford for self study ?
As someone who learned almost all of his math through self study, this is one of the things that truly frustrates me, and i don’t see any reason for this other than profit for publishers, or an attempt to force unmotivated students to get their grade (they’ll forget the content anyway because you can’t force motivation). Many of us understand the frustration faced by so many students when they are put through the needless, gratuitous suffering of reinventing the wheel that was a product of centuries of patient trial and error presented as if it is “trivial” by someone who has been mulling over it for decades. One of my favourite math book authors: the late and great Robert Ash was an engineer before he became a mathematician. I believe he understood this frustration and dedicated the rest of his career to writing undergraduate and graduate level math textbooks with detailed solutions to all of the exercises. In his paper “Remarks on Expository Writing in Mathematics”, he recommends including solutions to exercises and says: “I trust my readers to decide which barriers they will attempt to leap over and which obstacles they will walk around. This often invites the objection that I am spoon-feeding my readers. My reply is that I would love to be spoon-fed class field theory, if only it were possible. Abstract mathematics is difficult enough without introducing gratuitous roadblocks.”
I always hated those "The proof if this is so trivial it will be left as an exercise to the student ..." One professor I had used to assign those as lessons for the students to teach. The one I got took me half a semester to solve and when done, the solution was 27 pages long. Might be trivial on the rarified air of PhD professors, but not so trivial for an undergrad. Turns out also that I was the first student he'd ever had, including himself to derive that solution.
As a self learner I think having answers or solutions manuals are very useful.
Having the answers. Especially with the step by step solutions. Can be great. Sometimes it's just a tiny part of the problem that is tripping you up. I feel lucky that we can basically look up answers for any book these days. Banging your head against the wall is good. But if you do enough problems, you can achieve the same result with less pain.
Don't bang your head against any wall.It's bad for your frontal cortex and doesn't help with any problem.Mathbooks with answers are better.If you're stuck and you want to practice you're lost without solutions.And it's not true you're not making progress looking at solutions.Sometimes, you need them to even find the questions you're looking for and when the problem is solved, you can try a similar one,this time on your own.
"The basic answer is that humans are wired to take the easy way". Yes that's why we're learning advanced math that 99.9% of people will never do or understand(because it's so challenging and takes tremendous amounts of work and labor), because we mathematicians like cheating and memorizing thousands upon thousands of answers. /s
Are you talking about students trying to pass a class or mathematicians learning math? How about musicians, body builders, doctors... Saying humans simply choose the easy way abstracts away so much humanity it's sickening. People like to work hard and push themselves. There's more evolutionary arguments for that way of doing things then simply sitting down meditating all day; we would have died in the forests if we were wired to be lazy. Humans are hard workers. The lazy ones who wish to do nothing other than cheat themselves are broken humans, who maybe are just responding to a broken system.. It's complicated.
Anyway, not including answers makes self study almost impossible unless you have access to other resources(how can you prove something by rigorous mathematical standards if you have never seen examples of proofs before? You would have to create logic and set theory from scratch, proof by contradiction, double negation, negation introduction, etc..)... Can you imagine trying to learn music simply through trial and error? How about body building? How about anything.
Sure it's possible, but removing answers removes a critical element of teaching. The demonstration of satisfactory techniques. It could take you a thousand hours and lots of muscle strain to hold a guitar correctly or you could learn in an hour with a music teacher. Is that cheating? Should the student have to learn everything?
It just makes no sense unless you believe the vast majority of people learning calculus would rather not learn it and just cheat. This doesn't click with all the millions of man hours and resources invested by people wanting to actually learn math. highering tutors, subscribing to math channels, going to higher education, buying countless books...
Are you saying engineers don't realy want to be engineers? Scientists don't really want to do scientific work? That humans are just cheaters? Ok math sorcerer, let me ask you, when you read a math book, if it had answers, would you never do any of the work? We both know the answer. You would do the work, because we humans aren't wired to just do easy things.
*edit* this also adds to math anxiety. The idea that students must do everything themselves, that they shouldn't be taught correct procedures. Students must figure everything out and do everything.. If that's the case, is watching your channel cheating as you're helping people find good books to read. Shouldn't the student read every math book and figure out the best books on their own? Answers shouldn't be given to math students, we must figure everything out on our own right?
Anyway, I love this channel and will always be supportive. Thank you Math Sorcerer.
It's the numerical answers that are needed so you can check you've got the sum right right. Not so necessary for a proof / derivation as the answer (the thing you're trying to prove / derive) is stated in the problem. From a UK perspective, the only thing that counts is the end of year exam, not progressive assignment grades, so there is no incentive just to copy an answer to get a good homework grade. (in my experience, UK books normally provide all the numerical answers). Also why sell separate solutions manuals to instructors? If they are so f***ing clever, the instructors shouldn't need them.
That's interesting. I have some older UK books and they do have answers to the numerical exercises. I guess this is common throughout. Thanks for this comment:)
IT IS BETTER TO HAVE FULL SOLUTIONS. If you are properly incentivized, the problem of mindlessly copying the answers to get homework points is a non-issue. Test prep materials (like for GRE, LSAT, GMAT) or professional certification exams (e.g., actuarial) pretty much always include full solutions for every problem. This is because it's more effective and efficient. Without solutions you can get stuck on a problem for hours because you're missing something extremely basic, you're misunderstanding the question, etc. There's is not much benefit in my opinion to spinning your wheels for hours because you misunderstood something. With a solution, often these things can be cleared up immediately. With solutions, you still get stuck, but it will usually be on things that you legitimately do not comprehend. It's better to put in the effort at the frontiers of your comprehension.
The argument I give why all solutions should be given is that if a person computes a problem, or a concept, and they think they're right even though they're wrong then they've given themselves more harm by forming a conduct of behavior of approaching that problem or concept. When someone is learning, feedback is crucial. The behavior/habit of approach can be corrected immediately.
I absolutely hate it when the book has no answers. The philosophy is framed as a way of having students adhere to the path of "struggle", and guarantee that they suffer through the problem to the very end, but I don't care for such exercises. I prefer to learn something and move on, so I only do the exercises to see if I can do it right. If there aren't any answers, then that philosophy, instead of setting me hard at work, just leads me to immediately quit. I will not work for hours without promise of results; I learn best when I'm told the answers.
Brazilian elementary math books usually have all the answers
There's a calculus book collection from a brazilian author called Guidorizzi which have all the answers as well, iirc.
Oh interesting. Thanks for sharing this.
The reason also applies to programming as well. By struggling through difficult subjects you slowly gain insight into the subject at hand. If you stick with it, you reach a point where you finally get "it". When you learn this way you will understand the meaning of a "eureka moment". Realizing you can teach yourself how to learn difficult subjects is one of the most satisfying accomplishments you can make in your journey.
One key difference is immediate feedback from the compiler or interpreter. That immediate feedback is vital for learning. It will tell you where the error is and, depending on the language and tooling, provide hints on how to fix it. No such feedback with math texts.
At CSULB back in 1986 we had a guest speaker, Edwin Hewitt from Univ. of Washington. He was the author of Hewitt and Stromberg's Real Analysis book.
I asked him if there was a solution manual and he said, "No," He also said that he and Professor Stromberg had at one time worked out all the problems, some of which were real bears that a lot of the professors teaching real analysis would have trouble with and were designed that way.
Moral of the story is don't assign problems you can't workout yourself.
Anyway, he signed my book. It's an excellent book, but at the time, I only worked out the examples, and looked to other sources for exercises.
I definitely understand from the perspective of the book being written for teachers to instruct students why answers shouldn't be available in the book. That is why I always get supplementary books and material which have similar, complete problem solutions for all of my courses to facilitate learning the material and being able to see if my answer is correct only after giving it my sincerest, best effort. A lot of popular college books these days have very poorly written and demonstrated examples which create a lot of ambiguity which other material can clear up. Working similar problems and having the means to see you did it correctly is an essential part of the learning process, much like having a personal tutor show you where you went wrong without the costly expense of hiring one. A person learns more from one caught mistake than by 100 correctly solved problems. Nothing is more detrimental to your study of mathematics than proceeding further in the field on an incorrect foundation. Submitting an assignment and hoping for the best without having the means of checking your work is my definition of reckless. When you care about your GPA, this approach really helps you makes sure you have the correct understanding of the material.
You could have ended the video right at “we’re never going to reach a point in history where we have all the answers.” A philosophical explanation!
i'm 28, so i'm an older student in part-time uni that teaches via distant learning but most of my time is spent self-studying, the reason i prefer books that have worked solutions online or elsewhere is because i usually need a lot more worked examples to get my head around the concept. But as most of you already know being in uni feels like you need to rush everything.
i am very much looking forward to graduating just so i can pick a textbook and not be under any time constraints to finish it and work through the problems.
My linear algebra professor chose a textbook with all the odd-numbered answers in the back of the book. His homework assignments would be "Do all the even-numbered exercises at the end of the chapter". Usually each chapter would have 100+ exercises, so that meant 50+ exercises per homework assignment, several times per week. It was tedious, but you got good at linear algebra! Often if you didn't get a concept, it was nice to be able to work a few of the odd-numbered problems, verify that you're correct by checking the back, and then applying the same logic / template to the even-numbered problems.
You have forgotten one possibility: the author doesn't know the answers 😀
i think we learn by quantity, and learn from sheer amount of datasets,
the more the problems, the more the example, the better we get.
but having no answer to the question meaning that we have an unecessary friction when learning,
especially when you're learning by yourself, the last thing that you want is to mislead yourself into a wrong answer, and thinking you're right.
i think it's important to have the answer for things, and also have the actual problem solving process,
so that we can have a libraries of problem solving technique.
or else we have to reinvent the wheel each and everytime we learn about the new problem.
i want to be able to see the solution, dissecting the technique, and learn from it, without having to solve it from zero everytime.
and i always have this problem where i can't improve my skill fast enough because there are no abundance
of Math problem, because people treat a math problem like it's a sacred things
As a research science mathematics student, I want to ideally make every assignment, from every chapter, from the particular book given at the particular class. Also the exams are being viewed by me not as self-testing but as a quick method to prove everything I should already know, is being understood, because I studied the complete structures of every and all assignments belonging to the examined questions.
Also I argue that if one is going to write a mathematics book; and one would like to make assignments accordingly, that one should check if he can make those assignments according to the theory one has provided himself. A true mathematician wouldn’t be too lazy to do such thing.
Furthermore, mathematics professors must get rid of their fear. Real mathematics students will want to do complex mathematics by understanding, not by copying without comprehension.
Regarding the "nonsense" about the so called struggle. The only reason here could be that people are just jealous. Because one did so long to comprehend something, one apparently cannot bear that someone else can then with the knowledge provided learn the same thing in significantly shorter time, therefore getting ahead of the one who provided the initial solution in the first place. As long as this poor behaviour exists, it will take years to build bridges.
As they say in the US Seal Teams: “Nothing is done alone.” Therefore as long as answers aren’t provided because some ego issue of the writer of such book, real motivated students wouldn’t be able to study the particular structures of the solutions. Only to test the theory, but that would be half work, therefore a waste of time.
I argue, books should have solutions for all practice exercises, but separate solutions for the exam exercises; only provided to universities to prevent exam answers being spread.
I hereby rest my case. For people who are learning mathematics and are in their starting stages, I strongly advise to study the structures of the solutions, not necessarily the fastest way to the answer, but the way that you yourself can comprehend and therefore makes you able to perform maximally to achieve total mission success in explaining your own solution, while giving the right answer with it.
One of the main reasons I believe math books should have answers is very well explained in Veritasium's video about what it takes to be an expert, awesome video if you haven't seen it. One of the points he made that its important for becoming an expert is timely feedback, if you don't get feedback, you don't improve. If a book has no answers, you don't get feedback on what you did, and even if you assume the student is part of a classroom and not self taught, it can still take weeks to get a proper feedback on your assignment. Because of that, books without answers go against what is needed for you to properly learn, not only maths, but any field of knowledge, which is odd for a book that's supposed to teach you something.
I had to teach myself from a College Algebra level to Discrete Math & proofs. The math industry is NOT setup for self-learners, or those at their own pace. I do not learn well taking 32 weeks to get through a book in class that I can do in a month on my own. Tutors? Couldn't find any outside any specific curriculum or not using the same book they themselves learned from. Answers? Not easy to find any. Online help? No one wants to "to your homework for you". Online software to help with it? Only accompanies the book and costs as much as a semester and not always purchasable outside a specific university class. Google for answers? Not even ChatGPT gives good answers on proof type questions (given theirs so few available answers, it has little info to learn from just like us human counterparts).
I had to press thru and I"m not even sure all my methodology and answers are fully correct. But its the best we can do without enrolling. One day I will hope to change this for others.
I personally don't agree with the first reasoning. Anyone who wants to learn and cares about learning won't take that approach. I don't know if it is only me but I genuinely enjoy learning and doing these problems , I would bang my head against the wall to figure it out regardless of if the solutions are given or not. I cannot tell you as a self learner the amount of confidence I gain when I solve a problem, check it and it is correct. Math profs are failing to understand how important that confidence gain is. There are times when I choose difficult problems to do but when no answers are provided I choose not to do them
I studied math and physics at Moscow Institute of Physics and Technology (legendary Soviet university where major Soviet Nobel laureates taught students, one of the strongest of its time). There were special teachers who taught only how to solve problems based on the lecturers which professors read. These people were really good, many ofbthem wrote own problem books of different level of difficulty. Tradition was (and still is) that they have answers. Homework often was not graded, only tests and exams. I think it all depends on traditions of teaching schools the authors and students are in. But thank you for input, very interesting
Wow that is so cool!
I think as long as the student attempts every problem they should get credit so at least it will encourage the students to try. If they grade for correct answer it will encourage the student to copy because grades are so important.
Some questions in advanced books are nearly impossible for average intelligent students. However, usually, there are some geniuses on Stack Exchange who can give some hints if you show your work to prove you have done your best. Overall, having full solutions for every exercise might not be a very good idea because you can’t learn math without challenging yourself, but having some hints for difficult questions at the end of the book would be very helpful.
I don't usually do this, but I'd leave a comment to help this video get more views, your channel is a gem for a self-studying enthusiast
Students can get misconceptions from answers, too. If you look at an answer, it matches yours, and you think, “Great; I was 100% right and so therefore I can stop thinking about this now,” then you’re making an invalid deduction. It’s not too hard to come to a correct answer while missing key insights because you’re leveraging strong heuristics, etc. There are also situations in which you’re going to want to continue thinking about a problem even after you’ve solved it.
I’m one of those people who *rarely* looks at answers. I’ll struggle with certain problems for weeks if they feel important enough. I think that’s been extremely valuable because the math I *do* know, I can teach from the ground up and explain every detail of. I have my own mental models I’ve constructed which I can explain to my students, and I feel like I have a personal relationship with the material. You simply don’t get that if you treat working problems in a highly regimented and transactional manner. As a result, I’m always very skeptical when people want to look at answers habitually after they’ve solved a problem-or especially if it’s even before they’ve solved the problem!
HOWEVER, those few times I have sought solutions, I felt like I got something valuable out of that, as well. Especially when the content of the problem is important in some way, and the content of the solution had a really new idea that I had never considered. Because it’s so rare for me to look at answers, these few cases where I’ve seen answers stick out to me in my memory, and those answers are the ones I tend to remember best!
Overall, my impression is that people deeply entrenched either camp “answers” vs “no answers” would probably do better to move closer to the other side. If you look at answers semi-regularly (say, significantly more than 2% of the time), then you’d probably benefit from asking yourself some tough questions. On the other hand, if you essentially never look at answers (say, significantly less than 0.1% of the time), then you’re either missing out or are some kind of outlier (either genius or incompetent). I don’t know what the optimal balance is, but I bet it varies from person to person.
I think it’s probably good for books not to have answers. If you really want to find an answer, then you probably can find it online (or, if you’re so lucky to have access, by consulting an expert). I think it’s WAY more common for people to be inclined to be too eager to look at answers than to stubbornly refuse to look at answers (despite that not being so uncommon that it’s shocking whenever it is the case). As a result, adding some obstacles to looking at answers is probably good for the majority of people (even if it makes the “no answers” evangelicals more annoying).
As someone who has struggled self learning from books, I think I would be right in saying it is a total no brainer to have worked solutions to all the problems, in fact it is the only way you will learn. Your reasons why they don't have answers are correct, but, alas they are all excuses. First point, learning should be made as easy as possible, banging your head on the wall means there is a failing in understanding, bad thing, you may give up the subject. In easier areas teachers are constantly saying you will not get good grades unless you show full working. That is the beauty of worked solutions it shows you how to lay the problem out. Point two I think is the real reason, teachers want to use the even numbered problems as homework. The solution is easy, the teacher gets their own set of problems to give as home work. Point three it's a lot of work giving worked solutions to the problems. Yes but loads of keener students (for a bit of cash) could do that for you, also if you had a little team on the job then they could cross check the answers. There is another reason. Worked solutions take up a lot of space. In fact you may well need a separate book, something publishers probably don't like, but hell it's worth it!
I think there is tremendous value in "banging your head against the wall". At a certain point though you need to be able to check if you've done something correctly or not. At least include all the answers on the publisher's website (for free). That way there's an extra step so you won't just be tempted to immediately look up the solution without putting some effort in first.
@@jacobjones8131 There is no point in studying if you are going straight to the solution. I think a standard method of self learning is to go through all the problems, using the examples for the procedure. When you have done as much as you can then mark yourself using the answers. This is why worked solutions are so important as you can tell where mistakes havebeen made.
If you get a solution to a differential equation for instance, just put it in the equation to see whether you get 0 = 0 or 1 = 1. The point is to learn to check it by yourself as finishing the exercise.
If the exercise is like solving the equation, linear system or congruence, derivative, integral, etc, I believe that having the answers are important to study to know if they have success
Exercises about proving a statement should have not the solution, but some hint like "use Theorem 3.1".
I remember that when I begin to study Real Analysis, some lists of exercises I couldn't solve, for example, Exercise 3. After many days trying it, I asked help. My teacher said that I had to use Exercise 2. For the very first time in my life, I realized that I could use previous exercises to solve others. Unfortunately the book did not suggest that. I also thing very unfair when you have to use, for example, Exercise 5 from Chapter 3 to solve Exercise 10 of chapter 6, without this suggestion
For me answers are crucial in a book. And even better if they explain it a bit. Otherwise you are banging your head and you don't advance a little bit. Because you just don't find it. To me you also learn far more because you can see where you went wrong if you have an answer with some explanation and you tried and tried to come up with a solution. I am self learning by the way.
I will never understand educators who prescribe “struggling” in a vacuum. Feedback is the most important aspect in learning something. You need a way to assess your own understanding. Plenty of times I thought I had a good grasp on a concept only to learn later that it was built on sand. Also, I don’t subscribe to the idea that educators should “save learners from themselves.” The folks looking for the easy way out are not learners and it ends up punishing those “struggling” to learn. Those who really want to learn won’t be satisfied with just the answer. Many times, I’ve sat with an answer continuing to work through the problem because I wanted to understand. I almost didn’t continue with my Math career because I routinely encountered educators who weren’t willing to provide solutions. There are many ways to “bang your head” with this stuff. It’s very frustrating when educators pander to the wrong audience. It’s incredibly misguided and unfortunate. The discipline needs to reevaluate this approach.
I had plenty of struggle with math despite having the answers. My process was usually to try and solve, check if my solution was correct, then try to find out why my solution is wildly different.
Imagine you spent hours, possibly days, on a problem and you know you're so close to succeeding. But you made a mistake somewhere and you cannot find where, or you wrote almost the entire proof but you're stuck right at the end.
One cannot describe the relief when you read the answer and you finally, finally understand your mistake.
Not everyone has a teacher to help him. But everyone has the right to know the answer.
Thank you for sharing those thoughts. I think it goes to personality. I know math students who just grab answers. For some it's a matter of pride to fight endlessly on one problem until it's solved, before moving on. Not including answers keeps the bar for success high without question.
A fourth possible reason why there are no (or sparse) solutions:
Particularly with proof heavy courses, the goal often isn't to get an answer but to show why something IS the answer. There are multiple different ways to get a proof, and part of learning math is learning what's the best way for YOU to look at things. If I give you a proof of something, you're going to understand how I did it, but not how YOU should do it.
Abbott's Understanding Analysis first edition has answers (online but published by Abbott) to every single problem!
I wish that Munkres would have more examples. Sometimes I'm trying to work out an example on my own, and then I look at the book to see that that specific example problem has absolutely NO solution, so I don't have a Scooby Doo if I'm on the right track. So, I don't ever really feel comfortable with my homework if I can't work out the examples and be confident in that.
I respect the opinion of the Math Sorcerer, but I disagree. Not providing the answers to the problems is like having to reinvent the wheel over and over again and can actually hinder progress. Nothing about math is absolute or universal, not in time or space. We have learnt and lost mathematics at different points in our history. Case in point, we do not count as they did in ancient Egypt and we do not use abacus's as they did in ancient China. If we are to move forward with quantum computing we need to look to the future of mathematics and not struggle with subjects that were more relevant to the past of mathematics. Give the students the answers, which is how students learned during the Renaissance, and then use that knowledge to expand the discipline. That is just my opinion.
Reinventing the wheel over and over is a huge waste of time. You can be taught how the wheel was made, and then with that knowledge invent a better car. Each persons life shouldn't simply be doing everything all by themselves. Also banging your head against the wall is one way to learn, but it's surely not optimal. Answers are critical to the learning process. Can you imagine trying to teach a class without showing answers to any problem? The class would just be a list of problems that need to be solved. What's the purpose of this? I'm sure it's great for tenored math professors who love spending 80 hours a week lost in math, but what about those have other goals and purposes?
@@ai_serf I agree. Newton said, "If I have seen further it is because I have stood on the shoulders of giants." Newton was definitely making the point that he didn't "figure things out for himself."
I'm currently writing a textbook on Asymptotic Theory. I've decided to create two versions, one with full solutions (for instructors) and one without (for students). In LaTeX, that's easy to do, I have one LaTeX file with a switch set that includes the solutions, and another file with it unset. There are about 100 questions, and the worked solutions add an extra 80+ pages to the length of the book of around 450 pages). To be useful as an integrated part of a course, detailed solutions cannot be in the book. Equally, anyone wishing to mark students' work will need a set of solutions.
The questions I've created are usually of the form, "Use such-and-such method to show ..." and so the final result is given in the question. In writing a full set of solutions, it meant I had to ensure that the main body of the text was written in a way that could be used to answer the problems (in particular, the hypotheses of some theorem needed to be tweaked so as not to exclude the application in the problems). I found that helpful in developing the text.
In real life problem solving,
- you never have the solution before you found it yourself
- you have to first figure out with what lens to even analyse and with what methods to approach a problem because it doesn't appear in the immediate context of a segment of theory like it might in a book
- and sometimes you end up trying to solve a problem that doesn't even have a solution of the expected type.
So it's very useful to grow an awareness of what kind of mathematical environment a problem exists in. You know how to drive from point A to point B, but where is point A?
Yeah, one of the challenges with teaching is developing feedback that elicits the correct response from students and also teaching students to respond to feedback appropriately. Answers give students an idea of where they should be which is important but only if they then try to achieve the outcome on their own. With high school students I ask them to make notes on what their error was and how to fix it as well as attempting to prove the answer. In university I'd expect retrying and researching problems to be a given.
My PhD advisor wrote a chapter in a book and put some open questions in the end of the chapter. No answer was known and the student who solved those problems earned her PhD and based her career on the work she did in solving them.
A little late comment here, but I agree: It would not harm a sincere student's learning process to have math textbooks with answers, maybe even full solutions to every problem. Those mature and responsible enough to genuinely learn will not want to quickly look at those and will go through the joy of the struggle of discovering or creating solutions to problems. For example, I got a very good elementary algebra textbook by Robert Blitzer, an annotated instructor's edition with answers to both even and odd problems, but the answers were *right next to* the problems. It's nearly impossible not to get spoiled by that 😄so I bought another edition for students 😄 with answers to odd-numbered problems at the back of the book. By the way, I recommend that book by Blitzer, "Elementary Algebra for College Students," or any other elementary algebra book geared towards older learners. Thanks!
I won’t use a book that doesn’t have at least some answers. Feedback matters for self study. Having answers doesn’t really matter for courses either since an instructor should grade on the steps used to get the answer. Not just the answer. Of course I’ve heard arguments saying if you don’t get the correct answer then you shouldn’t get any credit at all. I don’t like that, but some do grade that way. This topic itself could make for an interesting video.
Excellent points:)
Pre internet this was a huge problem and discouraged a lot of people from moving forward. I personally didn’t really start to understand math until TH-cam resources started to really grow around 2011. This led to me eventually getting my degree nearly 25 years after high school. I am pretty sure with out videos by MIT, Khan Academy, PatrickJMT, Krista King etc. I may not have gotten far.
Now days if you look hard enough you will find the solutions online. So if I was to write a book especially a computational based one I probably wouldn’t include solutions out of laziness. There is basically a whole industry for providing math solutions so just let them sort it out.
I've been doing math with my 13 year old. I showed him a problem that I thought I had gotten correct and asked him to check it. He said, "Wait a minute" and took a picture of the problem with his phone, this plugged into an app that read it and worked it out in steps. Yep. That's right. He didn't even have to enter the equation into the thing.
@@MCJSA Yeah it’s pretty amazing there are a few apps that do that now. One day there will be ones that do it pretty consistently for proof based problems if there already isn’t .
chegg lol
I think it heavily depends on the task whether a solution even qualifies as a shortcut.
If it's a proof, you can't really present a solution without giving the whole answer away, so I see the argument there. But for most tasks a solution doesn't even shortcut anything, as you're typically graded on your calculation method.
I agree with your reasons, particularly #1, since I only ever had 1 math professor who graded the assigned work. I check my work on the odd problems, but it's frustrating when you just have to guess whether you've done a problem correctly or not. I think I'm learning, but it's hard to tell when I don't know whether I'm right or wrong on a solution. Now I'm at a point where I'm self-learning, and it's even worse since "cheating" is irrelevant.
There are like a dozen core proof techniques. Seeing them used can help; but the head banging is also needed.
Good math texts have the proof techniques demonstrated in the main text before the questions. Like, the lemmas and proofs not in the question section.
In that way the sample Q&A has already occurred. Just not in the question section.
Self Studying from books without answers can be very hard. I picked one such book for real analysis, by Stephen Abbot. It was hard. And it changed my life.
I discovered my true love for mathematics.
I've been reading Introduction to Inequalities. It has all of the answers, and I really appreciate it. Most of the exercises are so simple that I don't need to look. But there was one where I just got stumped. Once I saw the answer, I was able to reflect on why I had missed that, and how I could approach problems differently so the trick he used would occur to me. The next time a hard problem came up, I was able to apply those reflections and solve it myself. If the answer hadn't been there, I wouldn't have gotten that opportunity to learn.
Ciao!
Math books are not equipped with fully worked-out solutions because, in addition to taking a lot of time, other competing departments would get possession of new ways of solving problems that exercised the pioneers/teachers in that subject. That's true for the modern emerging theories though.
The most valid reason is the following.
If teachers provide worked-out solutions, one would merely learn by heart how to solve quantitative problems, and the exam would no longer be a valid test that measures how well one understood something. Rather, it would be a test to measure how good one is at learning by rote.
As you pointed out writing a math book is already a Herculean task, but writing excercise for a math book is just as much of an Herculean task. Many amazing book have no excercises and that is terrible. To the point of which they can become unreadable and even unapproachable. T_T
Solutions are kind of the least of the problems, it is the presence of excercises that makes for a great book. Where I come from we were not graded by the excercises on books so that reason doesn't strike me as much.
The reason I thought books stopped having solutions after high schools was because it is usually used to check if your solutions are correct, and at that level it becomes evident if your solution is correct or not. A reason the books, or professor's notes, on geometry in my first 2 years hadn't solutions (to be clear and to avoid confusion between different curricula they covered: linear algebra, affine geometry, euclidean geometry, projective geometry, hermitian geometry, conics, and little more) was because most solutions could be checked if correct.
Going forward some excercises forced me to write pages of calculations and calculating sigularities and stuff, so I can imagine as to not want to write them all.
As a side note, all the past exams with solutions were given to us, at least for these 2 courses of geometry, and many others, but I feel that this is a different story from what this video was about.
It's the guardian attitude. These folks want to stand over your shoulder and make sure you're "doing it the right way." Can you please just go? I've got this. Worry about your own education. Very frustrating for self-learners/lovers of learning.
Yes there are definitely people like this!! Not everyone is like this, but there are certainly plenty of people out there like this. Good comment thank you:)
When I took freshman calculus, we had to buy the text in the university bookstore. Next to the calculus text, there was the student's solution book (all odd numbers). The calculus text was thick and heavy (2 inches maybe?). To include the solution would just make it way too heavy to lug around.
I have a longer explanation as to why I dislike books that forego answers at the back, but my short response would be that: I don't bother with questions that don't have answers to them --they're dead to me.
Great reasons. Basically it's: No pain, no gain! You'll never become a mathematician by looking at the back of the book for answers without putting in a LOT of TIME & FOCUS trying to solve it yourself. This forces you to THINK on your own. The worse enemies of a math student (and everyone in general), are Lack of Time and Discipline.
If you're buying a book for self study, always try to get an instructor's edition or an "instructor's solutions manual". The main reasons these books don't come with all answers is to be appealing to colleges, so that they can assign homework that doesn't have solutions at the back. But for self study, answers are critical.
it allows the author to include unsolved problems to which no one knows the to answer (yet) and it likely forces one to develop confidence in one's own work
Outside of the US, the textbooks are for extra background/practice. Dont have time to spend on problems if you don't know whether you were correct or not. Even a final solution is fine, so that you can see, after doing the work in the middle, if you arrived at the correct solution.
I learned recursion by sitting in the dark over 3 nights for hours with nothing more than a hard, cold bench. I haven't been a morning person since.
From a practical perspective, the total number of pages in a book is a multiple of 4 (pages per sheet). The number of sheets in a signature (a stack of sheets) is also commonly an even number. Now the total pages is a multiple of 4 times sheets in a signature.
Each additional signature raises the overall cost of the book. The number of books that fit in standard-sized boxes is a factor.
Generally, the most daunting challenge for book authors is choosing what to eliminate to fit the allotted signatures. "How many pages do I devote to answers?" "Do I really need an index?" "Is my pet theorem actually relevant to the topic, and does it really need an entire chapter?"
Pro tip: Do NOT piss off your editor.
(:
Don't forget, too, that better quality paper is not only more expensive, but often both thicker and more dense, also leading to size and weight issues.
(:
@Tracchofyre
Candidate: "Your proofreading my thesis was more difficult an experience than defending it."
Me: "That's the idea."
(:
Solutions are generally available for download or for purchase for most books. In my opinion they are almost essential. BUT, it is is very important to struggle before looking. Like for an hour, minimum
How to know if your answer is right without the answers ?
My cryptanalysis professor self-published our text. It was great! He said he would give us hints on exercises during lectures, but that was it. We got no hints on our group projects (exams), and Appendix E: answers to selected exercises said, "If you think your cryptographic opponent will give you hints or tell you if you're getting warm, you're delusional."
Not having the solutions can be a problem because in order to learn, people need feedback.
I totally agree with the "struggling argument". When a teacher explains something in the blackboard it looks so easy that is not even worth the time. Having the answer (with the process) gives the fake sensation that it is easier than it is! But this is if it has the process. If it only includes a final result, could be okay to at least check if what you have done is okay or not.
I respectfully yet strongly disagree. Sure, if a student looks at the solution once or twice and then moves on, they are not going to learn anything. Yet, the student has to repeatedly look at the solution and keep trying to do it on their own until they can do it without having to look. But in my opinion and experience, it is absolutely vital to have the solution to fall back on when you're studying. You just have to be damn consistent.
Not to mention sometimes(actually quite often!) there's a typo in the problem which will become clear in the solution. Those are the worst and they happen very often
I think they should have all the answers, but just the final answers, because the teacher can always demand that the student show all the steps of their work. Knowing the solution won't help you with that, but at least it let's you know if your final answer is correct. That way, if it is correct, you can feel a sense of accomplishment right away instead of worrying needlessly that your answer may be wrong. Of course, this won't work with proofs, because the final answer IS all the steps.
Your comment near the beginning about needing to struggle to learn really resonates with me. There are many problems I encounter in my classes that get me locked up. But I find that after wrestling with the problem eventually I find the solution, and in the effort to find that solution the solutions to other problems become apparent.
There Is two other factors: 1. Sometimes the answer does not even make sense, like in proofs the answers to all the problems will end up using more space that the material. 2. Related to the previews one: each additional page costs money. 3. Some books have teacher editions, or complementary material that contains the answers. That is more at the calculus and precalculus level.
I was wondering about that recently and, in my opnion, not having answers for all the exercises is a bad thing, and a possibly solution for that is creating a latex project on github, with volunters writing the answers for an specific book, so that people could have all the answers and the math communtiy would get more united. The only downside is that you would need to learn git and github first (assuming people already know Latex).
For those who not are into computer science or etc : github is, basically, a platform where you can create open source projects, you and and the comunity can program with many programming languages (such as java and c++) or they can use markup programming languages, like HTML or Latex.
I am a music major and am studying Ebenezer Prout's "harmony" the reason I am studying such an old book over anything else is exactly because of this, he wrote his personal answer to every exercise, though, it's sold in a seperate book.
I find that on application based textbook, like engineering, the lack of answers is not a problem because you can get a "feel" for your answer. Whereas in more abstract based textbook, not having answers can be very difficult to learn, especially if you are self taught.
I would think the best way to go would be for problems that have a short numerical or symbolic solution, to just provide that solution. If it's a proof, give a very light sketch of the direction that the proof goes. In both cases, that should be enough for students who want to know whether they got the right answer to be able to determine that, but it wouldn't give the entire thing away. The sketch of the proof or just the final answer would not be sufficient if turned in for homework, because the teacher wants to see all the details that show understanding of the material in the course.
One of my favorite textbooks is Lee's "Introduction to Smooth Manifolds" and what's really funny about him is that he can sometimes be found on math Stack Exchange answering this very question for why his books do not contain solutions for the exercises. It's always funny to be looking up a solution for an exercise and then just finding the author of the book there haha
Thank you Math Sorcerer, you helped me with calculus 2, and I found you yet again, also amazing video man!
Great to hear!
Here's what I do wish: that older, long out of print texts would have their worked out solution manuals available online somewhere. The Larson calculus series has a wonderful website with worked out solutions to every edition they've published. It's great.
I would say it's all about the money. In order for a textbook to sell, it needs to be adopted by a lot of professors who will "force" it on their students by using it in their class. What professors are looking for is not only a clear explanation of the subject but also a lot of exercises of various degrees of difficulty that they will assign to their students. If the textbook author has published the solutions, the students will, in all likelihood, be able to find them and the textbook exercises are now useless for an official course at a university because it would require the professor to spend his own time to come up with some nice exercises for his students and, let's be honest, very few want to spend their time on that.
“Calculus” by Earl Swokowski! What a flashback; I had that same book in school. He had a separate solution manual with complete steps.
The hypothesis that answers are omitted so that students do not simply copy the answers on assignments makes far more sense. The notion that the best way to learn is to bang your head against the wall, never knowing whether you are closer or further away from a solution does not make any sense.
HI Sorcerer, love your channel! I've been following it since you started even though, as someone in his 50s, I just want to learn math as a hobby. It is my dream to get a basic math degree. I plan to do your Udemy courses when i get a base level of pre-requisite knowledge. I knew someone that edited for textbook companies years ago and they told me that reason that odd-numbered solutions are generally given, is for a couple reasons: 1) It allows the teacher to use the even-numbered for testing and assessment, and 2) because a separate instructors manual and/or solutions manual could be also sold.
Are books the best way to learn math , Or are there other ways to achieve the same purpose ? and what's your opinion About Openstax math books ؟
Another reason I could see being put forward for not having solutions to every problem is that it helps prepare students for situations in which they're working on problems for which the solution is not known. You have to work on your solution until you're convinced that it's correct.
I don't know how much I agree with this, but I know a lot of people who might say something like this.
My biggest pet peeve is when they show the singular answer but no solution. So if I got the right answer with using the wrong method, then I may think that’s right and do it on an exam. And most of the time professors just mark a answer wrong but don’t tell you what’s wrong about your logic. And most of the time it’s not even a professor grading; it’s a TA. And they most of the time just look for the correct value. It’s just so frustrating when that happens. Especially when you get to higher level classes. And we’re paying for these classes.
What's fundamental is that most problems with no solutions already have it in how they pose the question e.g. "prove that..." and I think that is REALLY important, you know where you are getting to.
you could do little mini series videos/thin books (chapter or 2) explaining text book problems. step by step expounding various problems in detail, giving reasons for certain approaches. working with the authors and publishers of each book. it'd make money for both parties, help struggling students pass and from seeing half million subscribers...
be popular. and a 100 page student guidebook would be easier than a 800 page text to approach writing. I can see them becoming a cult classic like the MIR series.
I absolutely adore what you do on this channel. I'm a mathematics student and an aspiring teacher and this kind of content is so valuable to understanding what intent to have when teaching. Thank you
I was working through an older text, Calculus, Analytic Geometry and Trig. All the problems were word problems and no answers were included.I don't remember the name, but in the front matter the author said, " In the world all applications of math are word problems". He never stated why there were no answers, and I struggled with the rigor. I never finished the book and am actively searching for it. It was solid yellow. I'm assuming, you can check your answer by plugging in values to arrive at the initial problem.
I never copied the answers, but I would use them to check my work if I wasn't sure about it and knew the answers were provided. My interest was mainly in seeing if I solved the problem correctly -- if not, I probably would have got an entirely different answer.
So when I taught math, I was shocked by how many people just copied down the answers.
One of the more talented math students I ever knew would never completely solve the problems. He would take it down to where all he had left was arithmetic and he would stop there.
I asked him about it one day and he said that there was no need to do them. What was important, he said, was in doing the math to get to those final calculations. And, of course, in many of our advanced courses, there were usually no final calculations anyway.
You should make a list with books on every level and topic with full solutions.
I am sure that such a video would do well. Plus you can link them in the description.