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.

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    $\begingroup$ 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." $\endgroup$ – Namaste Jun 23 at 17:09
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    $\begingroup$ 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 Brander Jun 23 at 19:37
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    $\begingroup$ 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 Jun 24 at 13:20
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    $\begingroup$ 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. $\endgroup$ – Chris Cunningham Jun 24 at 23:46
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    $\begingroup$ 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 Jun 25 at 18:31

Because in real life, real questions rarely come with ready-made solutions to preview.

Hence, the norm I've seen, in early undergraduate classes, of texts providing answers to every odd (or every even), numbered question, but not both. You can test your understanding against some odd questions (or even), and check the corresponding answer to see if you're on track with your understanding.

But being assigned only exercises with answers, directs students to go to the provided solution from a text, and then copying the answer, does not pose a great learning opportunity.

Each student needs to engage in exercises for which they are not provided prescriptive answers to copy. Those exercises become the real learning experience. It's up to you to work it through, and offer a solution/proof reflecting your understanding. Or to seek out help. That's the only real measure of a student's understanding.

What you have wrong here, is your claim "if you don't know how to solve something right away then you need the answer to guide you."

That's entirely wrong. Solutions to some exercises don't always come right away, and those exercises are needed. Students need to learn perseverance, practice, patience with repeated alternate approaches, etc.. If every math exercise for every student was something students know "right away" how to solve, they wouldn't be learning anything new.

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    $\begingroup$ In real life you also have google and whatnot... "copying the answer, does not pose a great learning opportunity." I already addressed that, so I take it you disagree that it's the student's responsibility to work through the exercises rather than just peeking at the answer? "Right away" was a poor choice of mine, nonetheless the point was that the effort needn't be fruitful. $\endgroup$ – Erik Jun 23 at 19:06
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    $\begingroup$ I merely suggest that any informed professor or TA assigning problems from a text is not going to assign only problems with answers provided in the text. You're free to research and learn more to be able to answer assigned exercises which are not answered by the text. But where did you develop the expectation that all challenges you will encounter in your life must necessarily have solutions sets available for you to refer to? $\endgroup$ – Namaste Jun 23 at 20:44

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.

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    $\begingroup$ 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 Jun 23 at 19:20
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    $\begingroup$ @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 Jun 23 at 20:03
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    $\begingroup$ 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 Jun 24 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 Jun 24 at 22:49
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    $\begingroup$ @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 Jun 24 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.


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