Creating Sample Solutions to Math Problems

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  • What is the copyright position on writing a book of worked solutions to problems in a maths book.
    There would be minimum reference if any to the question and the book would explaint o pupils how to work out the answers.
    Any help on this matter would be appreciated.
  • I'm not sure what type of "maths book" you mean. Are you talking about a commericial text book usually ordered from the publisher or a trade book bought at a retail store? I seems to me the purpose of a math text book is what you intend to do. A bit more clarification would be helpful. Thanks.
  • I think what you mean is taking a list of math problems from a textbook and providing the answers along with an explanation of how they are solved. I believe this would violate copyright for several reasons. Once, the publisher probably already provides this in either a teacher's edition or a solutions manual. And if they don't, they still have the right to.

    If you want to make up your own problems that follow along with the subject in a chapter and provide solutions and explanations, I think this would be fine. Math problems are pretty factual, but I think providing answers in order to a set of problems created by someone else would be a violation.

    It's possible there is other reasoning behind this--anyone else?
  • Numbers and basic math problems with only numbers and symbols probably aren't protected by copyright. Word problems would be, as would any descriptive text. However, the selection and arrangement of even numerical math problems could be protected by copyright. So, it is possible that this solutions book would be a derivative work.
  • Many thanks for your replies

    Just to clarify the situation

    I have a maths text book that contains a series of exercises, for which only the solution is given in the back of the book.

    For each exercise I write my own detailed solutions to each of the questions for my pupils, is this illegal then?

    Could I write my own book of worked solutions to the problems? It is possible for example that a problem could be worked out in many ways and by providing actual workings for the solutions and not just the answers would clearly be really beneficial.

    Just a few further quereis that have arisen from a few comments made;

    firstly I am assuming you cannot copyright a maths question e.g. 23 + 37 etc... However somebody suggested a sequence of problems could be copyrighted. So lets say I write a book that mirrors the exercises in the textbook my students are working from (e.g. Q1. Solve 2x + 1 = 5 is in the text book and I write Q1. 2x + 3 = 7 and so on) would I be vilolating copyright then if my questions were designed to match up with the text book questions, so if they followed my solution to my problem, they could easily generate the solution to the problem in the textbook.

    I am really grateful for the help people have provided so far, but just need clarification if possible on the above points if anyone knows.

  • My initial reaction is that either of these would be okay. COValle is right in that while basic numerical math problems aren't copyrighted, the sequence or selection might be. I think you could argue that your work would be transformative enough to not violate their copyright. They might argue derivative, but I would feel confident in doing this.
    As for making your own that mirrors the text book, I believe there is no problem with this. As a teaching tool it is even better than just using the book problems and I don't see a copyright problem.

    My nonlegal opinion!
  • Many thanks for your reply LW. I am very grateful for the information.
    Could I just clarify that you think it would be okay for me to produce my OWN worked solutions to problems for pupils.

    Like I said before it would seem unfair that due to copyright the only way the answer to a maths problem could be explained is by the people who have written the original textbook.

    My intention would be to put the worked solutions I have written into a book and/or maybe on my own website.

    Is there anyway I could get a definitive answer to this problem?

    Cheers. I really appreciate all the help so far.
  • I'm afraid there is not definitive answers pretty much anywhere in copyright. Each case is analyzed to the best of your (or your lawyers!) ability and you reach a conclusion based on your reasoning. As I said earlier, the copyright holder may reach a different conclusion. In my reasoning, your use would be okay, but this certainly doesn't have any legal weight and even if I gave you a definitive yes in my opinion, that doesn't preclude the owner from challenging your work.
    You would be on pretty solid ground with your second option of making your own problems that parallel the textbook. I don't see how this could be considered a derivative work at all.
  • This response is certainly no more 'definitive' than the valuable comments already posted for this thread, but I'm throwing my 2 cents in anyway! The example I offer here might further suggest that creating parallel, but different problems for your use might be the more prudent approach.

    As a parent of a rising high school senior, I've noticed that the SAT prep books for sale out there all contain extremely similar but not identfical problems, each using nearly identical formats, and that NONE of them contain actual SAT questions except for the prep guide produced by College Board, the maker the of the SAT. This leds me to wonder how content such as problems and answers--
    including the most straightforward, factual math problems -- could be protected under US Copyright Law. If they weren't protected, than why wouldn't Princeton Review, Kaplan, and all the other test prep publishers use the actual SAT questions in their guides?

    So I am thinking that problems and solutions are considered as 'consumable materials' -- items that are meant to be used and replaced regularly and not routinely copied. Consumables are specifically cited in the prohitibtions to educational fair use as follows:
    There shall be no copying of or from works intended to be “consumable” in the course of study or of teaching. These include workbooks, exercises, standardized tests and test booklets and answer sheets and like consumable material.
    (House report no. 94–1476)
  • Hello All Again

    Many thanks for people's previous replies on here. However I was talking to someone else about the issue of producing model solutions to exercises in a text book and he said that surely this is what pupils do everyday when the teacher asks them to complete an exercise. By completing an exercise, giving their method of solution - is this an infringement of copyright? Are teachers asking pupils everyday to break copyright rules? I thought this was an interesting point and just wondered if anyone else had an opinion.

    Many thanks for the help again
  • I really don't see a parallel issue here. Each student who purchased a copy of the textbook (or technically, purchased by the district usually for each student) is entitled to do whatever they want with their copy. That is the purpose--to teach and solve problems. That does not mean it is okay to solve the problems and copy it for everyone to see. And sell it to them. These students would not be violating any of the rights given to creators in copyright law.
  • LWilliamson

    Thanks for the response, I'm not sure I was entirely clear though.

    I have a class of 30 pupils. Each pupil has a copy of a maths book purchased for them by the school.

    I ask them to complete a specific exercise from the book as homework. They must show me (as I always request) full working out so that their method is totally clear and fully explained.

    (Here are a few situations that might well arise)

    Situation 1 - Utopia

    All pupils complete exercise for homework independently of each other. They all hand their completed work in before the deadline. Whilst they were completing their homework, I was preparing my own set of model solutions (for which I don't want to get paid) - but which I would like to share with the class via handouts or the intranet.

    The same thing happens each lesson until the end of the year. I have solutions now to each exercise, which I have decided to collate together as a book.

    Is there anything wrong with this? Is any copyright been broken?
    At what point would my solutions and explanations be breaking copyright?

    Situation 2 - More Realistic (but still exagerated a tiny bit)

    10 out of the 30 pupils complete the homework as requested. One pupil though decides he is going to do the solutions to the problems in the exercise and then sell copies on to the rest of the class, who don't want to spend hours doing the work. So this pupil produces his/her solutions and sells them on to their fellow pupils.

    Now is this pupil breaking copyrght or not? If not then what is the difference in someone like myself producing solutions to an exercise and then selling them on to be used by others - which is what my original post asked.

    Would it make any difference is this entreprising student gave the solutions away.

    Hope this makes what I mean a bit clearer.
  • Pencho:

    Whenever students create something in class, or complete an assignment, there is the potential that a student will create a new work (original, somewhat creative, fixed in a tangible medium). They are the authors and they are the copyright holders to the work they created.

    By selling their original problem solutions to other students is not an infringement of copyright if they have created a truly original, creative and fixed work.

    Selling the problems might be ethically wrong but it is not an infringement.

    The difficult point of this entire discussion is whether or not problem solutions are new works or if they are derivative works. Creating a derivative work is an infringement of copyright. A derivative work is a work based on the original work. But the original work must qualify for copyright protection or you cannot create a derivative work.

    If the original math problems are eligible for protection, then creating a new work based on the original would be an infringement.

    We have read that perhaps math problems (if numbers and facts) are not eligible for protection to begin with. If this is true, you have no problem.

    Do you plan to sell your math problems? There is an argument that the original author of the textbook used his creative abilities in developing, selecting and arranging the math problems in the text book. He was not just collecting facts but using pedagogic rationale for making up the math problems and putting them in a particular order.

    If the collection and arrangement of the problems in the book is worthy of copyright protection, you should be concerned about creating an unauthorized derivative work and putting it in the market place. Or putting in on the web.
    In the classroom, as a method of teaching the students math, your problem solutions are restricted to that educational experience and could be (probably is) fair use.

    It would help to know what you plan to do with your math problem solutions. It sounds like you want to market a work that could be used in conjunction with the original textbook. My opinion (having not seen the textbook) is that you would be infringing by creating a derivative work because the original work is protected by copyright.

    I think we have beaten this horse from every angle, and hopefully the remarks were helpful.

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