What is the point of exercises for which answers aren't provided? (That is to say, what is the pedagogical justification for such exercises? - Edit by someone other than original poster.)

Commentary behind the question by original poster:

Most if not all courses I've taken, math or otherwise, come with books with exercises where answers aren't always provided. What are students supposed to learn from such exercises?

I'm not necessarily talking about "difficult" exercises here. Indeed it's even worse when "exercise 1" doesn't come with an answer - the student has no way of confirming that he understood even that first paragraph of the entire book.

If you don't know how to solve something right away then you need the answer to guide you. The activity of trying and trying until you're completely exhausted and just have to give up, need not be fruitful, the approaches you tried may have been way off.

Yes, providing answers may cause some students to not bother taking the time to try on their own, but that's their responsibility.

  • 8
    $\begingroup$ This reads like an editorial, even a rant, at that. Your question in your first sentence reads like a rhetorical question, because you seem not interested in understanding "the point," because you go on to make it clear you have decided there is no "point." $\endgroup$
    – amWhy
    Commented Jun 23, 2019 at 17:09
  • 5
    $\begingroup$ The question is not opinion-based: Certainly there is a reason why half the exercises have answers, and someone might even know that. $\endgroup$
    – Tommi
    Commented Jun 23, 2019 at 19:37
  • 2
    $\begingroup$ For what it's worth, the core question here seems fine, and definitely on-topic, even if expressed somewhat baldly. If @Erik had simply put a dividing line between the question and the commentary would anyone have complained? $\endgroup$
    – kcrisman
    Commented Jun 24, 2019 at 13:20
  • 3
    $\begingroup$ Multiple people have deleted their own comments in this thread; as I result I've deleted some comments that were made obsolete, to avoid more confusion. $\endgroup$ Commented Jun 24, 2019 at 23:46
  • 4
    $\begingroup$ Not sure if this is worth a full answer, but I can say one thing that I think one benefit to working out a difficult problem and then not knowing if it was right or not, was that it helped me find resources and learn how to use them (WolframAlpha, Desmos, even Geogebra) in order to estimate the answer and see if the answer I came up with was reasonable. If I just had the answer, then I would confirm my answer or re-work the problem until it was right, and be none the wiser about some mathematical tools that were available. $\endgroup$
    – ruferd
    Commented Jun 25, 2019 at 18:31

2 Answers 2


To add to Namaste's answer above, two of the thing we're trying to teach in math is literacy and competency with the algorithms used to solve problems and the ability to solve problems for which you don't know the solution. The first is pretty well served by problems with solutions since you need to check your work. The second is the actual activity of doing mathematics, and is heavily informed by the first. And it's hard. But in order to do it, you need to practice doing it.

Basically, there's a whole list of skills that surround tackling an unknown problem:

  • Identifying the problem as being solvable with a certain kind of mathematics.
  • Planning out a logical series of steps and computations that will allow you to solve the problem.
  • Following those steps carefully and adjusting them as you run into issues.
  • Finally, analyzing the solution from a couple of different angles to know if you got something reasonable (The area under the curve is 4) or unreasonable (The area under the curve is -3).

All of these things take practice and confidence to do well. And they're hard, but just like playing guitar you can learn all the music theory and tabs you want but eventually you have to actually play. The fact that the doing is different skills is something you want to learn at home, not the first time you get on stage.

Mathematically, it is the difference between being able to do known analyses and being able to come up with a novel prediction mechanism, model, or formal description, use it to say something and be able to robustly describe your reasoning and back up the validity of your claims. To be clear, there's no reason to knock practical competency! A lot of engineering, computer modeling and practical statistics, uses known mathematics; the problems they deal with are interesting for different reasons and involve different kinds of novelty.

If you can follow your logic through an unknown exercise, check it three ways, and really understand your result, that is when you know a mathematical concept. Once you are comfortable with these steps, someone can hand you a novel problem and you can begin to reason through it.

  • 1
    $\begingroup$ I'd upvote this answer if I could. Now, you say that to get good at the second thing you have to practice. Certainly true, but how can you practice without being able to confirm your result? The act of trying to solve a math problem isn't like physical exercise where even if you don't make a new personal best you (your body) has/will still improve. $\endgroup$
    – Erik
    Commented Jun 23, 2019 at 19:20
  • 1
    $\begingroup$ @Erik I would say you start practicing on problems with answers, really trying to convince yourself that you have the answer before you check. Be explicit about your reasoning, and get used to finding your errors. Solve the problem multiple ways if you can (often you can use geometry, calculus or algebra for example), and see that you get consensus. Then use all of these tactics when you solve new problems. It's time consuming, but it also gets faster and you do slowly improve. It takes longer than say physical exercise, but that just means you should feel badass when you can do it. $\endgroup$
    – Nate Bade
    Commented Jun 23, 2019 at 20:03
  • 1
    $\begingroup$ The steps for tackling a problem are known as GFSA (matheducators.stackexchange.com/questions/14068/…), although this algorithm is more about how to write and present a solution in a clear readable fashion than about how to attack a problem. Still, a clear presentation helps find a solution. $\endgroup$
    – Rusty Core
    Commented Jun 24, 2019 at 18:52
  • $\begingroup$ @RustyCore This is really interesting, I haven't found many resources that teach how to think about open ended problems. Do you have any textual references for this? $\endgroup$
    – Nate Bade
    Commented Jun 24, 2019 at 22:49
  • 1
    $\begingroup$ @NateBade If I correctly understand the question, then How to Solve It by George Pólya is one of the classics. Also TRIZ / TIPS is somewhat similar. $\endgroup$
    – Rusty Core
    Commented Jun 24, 2019 at 23:02

Texbooks can be used for:

  1. Self-study. In this case they have a healthy dosage of theoretical material and a bunch of exercises, which have answers.
  2. Guided study, when the teacher explains theoretical material, shows how to solve exercises and then checks and grades homework and classwork. Some of these textbooks come in two flavors: for students and for teachers, in this case the teachers' version has answers and some additional guidance.
  3. Calling up a student to a whiteboard to solve an exercise in front of the class. Then the class is asked whether the exercise was solved correctly, and a discussion ensues. If an answer is known there will be no discussion.
  4. Tests. Split students into two or three or four groups and give each group an exercise that has no answer. Make sure that members of these groups are not able to talk to each other. Works best when the desks are arranged in rows.
  5. Search for older material. Most decent books build skills and knowledge step by step, so if you missed a step, you need to flip back several sections to find what you are missing. This implicitly forces you to figure out what exactly you are searching for, an important skill by itself.

In most cases the exercises that have no answers can be verified by the student himself: plug in the answer and verify that the equation holds true. If an exercise does not have a single best answer, which happens rarely in school math, then the student has to wait for the teacher to check and grade his homework. This waiting adds additional emotional tension, although most students don't care.

If you don't know how to solve something right away then you need the answer to guide you.

No, if you don't know how to solve something, then you need to work with the textbook, flip back several pages and see what has been explained there. Search engines condition students to sloppy thinking, sloppy phrasing, and do not require memorizing anything. Stay away from search engines, use your textbook.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.