# Solutions to exercises

I am teaching the exercise sessions for a 3rd year algebra course (intro to field theory, Galois theory and Algebraic geometry). The format of the course is as follows: for every 2 hour lecture by the prof, I teach a 2 hour exercise session where the students are supposed to actively work on exercises and I answer their questions. In practice this often means I give quite a few solutions on the blackboard, but not all of them. There is no homework for this course, just an exam.

Preparing for this course, I made latex notes containing the solutions of all the exercises. And so my question is: should I give the pdf of this file to the students?

On the one hand, having detailed, rigorous solutions for the exercises will help prepare the students for similar questions that might be on the exam.

But, on the other hand, I am worried that if the students have all the the solutions on hand, they will not learn the perseverance needed to struggle with a new problem (as might be presented on the exam).

Does anyone have some experience with this? Do students tend to do better or worse if they have access to all the solutions?

• Welcome to matheducators.SE. Presumably you are asking if you should give the solutions from a pedagogical point of view, and have discussed or will discuss with the lecturer before taking action, regardless of the answer here. – Tommi May 10 '19 at 13:38
• If possible, get your students to solve as many of the problems themselves instead of just giving the solutions. If they know that they will get a detailed PDF at the end of each class, that might not be the right way. On the other hand, of course, they should have all solutions in the end, so that is quite complicated to combine with the "no homework" rule. Try discussing it with your professor. – Dirk May 10 '19 at 14:37
• If you plan to have students work through a certain set of problems in a given class, what would be wrong with just waiting to give them the solutions to those problems until the next class period? If I have Solutions prepared, but I have alloted time for students to work on them in class, I always wait until later to give them the solutions. That way they use the time in class actually working. – Nick C May 10 '19 at 15:07
• What is the size of the class? – Ben Crowell May 10 '19 at 15:24
• Could you use your solutions to prepare useful hints for the solutions? The kind of thing I'm thinking of, if this were real analysis, would involving statements such as "using an $\frac{\epsilon}{3}$ argument one can show uniform convergence on $E$ to justify term-by-term differentiation to obtain a series whose partial sums are bounded by those of a certain geometric series, from which convergence follows", with lots of mature writing style phrases such as "passing to a subsequence if necessary" and "without loss of generality we may assume". – Dave L Renfro May 10 '19 at 16:23

If these are exercises from a published textbook, then it's probably self-deluding to imagine that the students don't have access to them already. Chegg et al. probably already have the solutions available to anyone willing to pay the monthly membership fee.

Your lead instructor has already chosen a philosophy and a set of rules. They've made the homework not count in the students' grades. That implicitly means that they're expecting the students to act like adults and do what is best educationally. These are upper-division math majors, after all. Even if you have doubts about their maturity and willingness to apply themselves and resist the temptation of looking at the solutions, your opinion about their maturity has already been preempted by the instructor.

Since there are only 15 students in the class, and presumably only some fraction of them will show up to the exercise sessions, it seems like you have the ability to interact with them one on one while they work on the exercises. This is an optimal situation, and one in which they can't really pretend not to be doing the work, if they actually aren't.

• Even for unpublished work the structure of Chegg, Course Hero etc. means that even within the same term the problems and possibly solutions if you deliver them asynchronously will be posted. Basically as students have access so does Chegg as it mobilizes the students as an army of spies. The only counter is to make new problems often and/or design traps to catch the Chegg-style cheaters who copy work without thinking. Of course, this is only necessary if you think the homework grade should count towards their course grade. – James S. Cook May 14 '19 at 13:16
• @JamesS.Cook: Even for unpublished work... I don't think it's quite as dire as you say. I've written some open-source physics textbooks that have garnered ~70 adoptions over the years, and only about 40% of my problems seem to have solutions available on Chegg. – Ben Crowell May 15 '19 at 20:28
• @JamesS.Cook: The only counter is to make new problems often and/or design traps to catch the Chegg-style cheaters who copy work without thinking. There are other counter-strategies. Simply reading students' papers has worked for me. For a problem that isn't just a trivial, cookbook-style computation, it's very easy to tell if multiple students have all copied the same answer from Chegg. Simply knowing that I will actually read their work (or at least some of it) seems to significantly decrease misuse of Chegg, vs instructors who only use an online homework system and never read papers. – Ben Crowell May 15 '19 at 20:30
• Certainly if you read homework papers I can see how that works. I would wager the other 60% of your problems have not been used with significant point values towards the grades of the adopters. But, you are right, it may not be universally as dire as I say. I want to be wrong on this. My data is mostly from online courses where I see the "quizzes" are posted on Chegg. I do think I have less cheating (offline) largely because of the human grading and/or oversight of my homework. It communicates the importance. It is also increasingly a luxury as so many adopt off the shelf online hwks. – James S. Cook May 16 '19 at 4:17

They probably can't finish all exercises in class. At the end of each chapter give then the solutions of the remaining chapters.

Having detailed solutions for the exercises will help the interested and hard working students progress and check their solutions with yours. As for the other students they will probably just read the solutions. Is this bad? Of course, but if they don't have the solutions they will probably just skip the remaining exercises or, at best, they will take the solutions of other students and read them. Needless to say these solution could have many mistakes. So it's better to give them the solutions at the end of each chapter, it'll help the weak and the hardworking students.

There's differences of opinions on this.

Liberal view:

I'm of the opinion that drill problems are for the benefit of the student and am used to using them on my own.

I have never had an issue with making use of them and/or deciding how long to struggle with something. I would note that even if you don't "get" the magic aha on your own, you can get some of the brain groove training effect by mechanically working the problem EVEN THOUGH you had to look at part/all of the answer/solution.

Feel like not providing answers removes a feedback mechanism. Also I disagree with providing the solutions days later. We are not computers. We are animals that learn from near term feedback. Behavioralism. We're closer to puppies than to Univac. More dog than God.

One other thing since you are NOT making this graded homework it is one more reason to be liberal on the solution availability.

Orthodox view:

Many paid teachers feel different. I have seen prefaces where authors said they removed some/all answers because buying committees preferred that. I'm sure that those espousing this view claim/think it is best. But I wonder if part of it is preservation of control for the teachers.

Since I disagree the conservative view, I probably don't explain or defend it thoroughly. But given board opinions you will get a lot of people backing up the orthodox view. More than the liberal. Want you to at least consider heresy.

• You make some good points, but please at least try make your answer objective. Now it sounds like nothing but a rant against the Orthodox view. – YiFan May 12 '19 at 22:45