What's the point of exercises without answers?

Question

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.)

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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.

Answer 1

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.

Answer 2

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.