Motivate students to work on exercises if solutions are provided
The way for a "not so good" student to become better is to work harder and solve more exercises. Reading a solution is not at all like finding a solution. Your test becomes one of memorization rather than skill if you test on exercises for which the students have already seen correct solutions.
I would suggest that if you provide solutions, rather than individual help on finding solutions and seeing where the students went wrong, that you post them after the test.
And, the key, post those answers with additional exercises for those who did poorly on the test to get some additional practice. If you want to incentivize them to solve those exercises, provide some bonus to the test scores for submitting correct solutions to the follow-up exercises.
But you have an additional problem in that your "practice sessions" are probably ineffective if they depend on students knowing that their supposed solutions are wrong and that they need help. Students have misconceptions often enough and need to be pushed past them. If they "solve" exercises but no one checks the solutions, then they are unlikely to see where they made a mistake. The exam only tells them that the are wrong, but may not indicate why and it probably won't help them dispel those misconceptions for the next try.
The combination of a bit of stick and a bit of carrot might help. You have the stick (the exam) but you are missing the carrot. Give points not just for success but for hard work, even if it is re-work.
I often learned lots of math by doing lots of exercises beyond what was required. It was only through that extra work that I was able to achieve insight. Luckily I was self motivated in those instances, but not everyone is. In another course the professor provided lots of external "motivation" through "pop quizzes" which we hated but knowing they were coming, forced us to be ready every day. Remember "flash cards" from early elementary school? Lots of repetition. Lots of feedback.
An option that I've never used, but might consider in your situation is to require that students who did below a certain mark on one weekly quiz, submit their work for review in the following week (or two). This might reduce the number of papers you need to give feedback on and also make it feel like a reward for the better students who don't need to submit. Or even, make submission optional to everyone. And comment on the papers you do get.
With TAs, have them comment on exercise papers and pass them to you for review.
The underlying concern that you raise seems to be how you can best engage students to become self-motivated in their own learning processes, especially "the not so good students", with the resources that you have at hand.
Perhaps you might first reframe this as a question back to the students. What will help you (students) become more self-motivated in your own learning processes? After all, what good will you ultimately do in your quest to give away anything that is either not desired, not understood, or not appreciated for its intent? You may also discover that what you believe about the desire of your students to be self-motivated in their own learning processes is a chimera compared to the desire of your students to just get a good grade in your class. As disappointing as the latter finding is, it can serve as a reality check to recalibrate where you might better spend your time to reach those students who sincerely want to do what is needed to learn and yet still be honest with those students who are just passing through your course for whatever reason.
One solution is to provide no solutions to the exercises at any point in time. Instead, give the exams with questions drawn directly from a portion of the exercises. The self-motivation is the commensurate statement "If you do all of these exercises, you will have done at least X% of the upcoming exam questions". Finally, you could address your desire to motivate self-learning by having an open-door policy to students who want to review the answers to their work.
Another solution is to provide all solutions to all exercises at some point. Realize that, in this day of internet and with the interactions that occur among students, the minute that you open this door for even one student or one portion of the class, you have essentially opened the door for all students at that moment and for all students in advance for all future offerings of your course. You can in this case address your desire to motivate self-learning by stating that exams will have a (smaller) Y% of the questions from the exercises and a corresponding Z% of questions that stay within the scope of the course but go beyond the exercises. Essentially, exam questions from the exercises would test how well students know the material (even if that is only rote memorization) while exam questions not from the exercises would be designed to test their mastery of (knowledge of, understanding of, and ability to apply) the material.
Finally, an intermediate solution is to give only a portion of the answers to the exercises. Again, with reference that students will have these next time around, when you decide this option, you may as well give the selected answers directly along with the exercises. Here, you can also balance how you distribute the questions for the exams. One potential advantage of this approach is that you can reserve some questions from the exercises that you do not give answers so that you can use them on exams. You might also simply give some exam questions that are already answered as exercises just to discover how many students will not even commit to studying what they are given to know it let alone studying to learn it.
In summary, each of the above three options has its own balance of resources and outcomes. I think that none of them are inherently wrong or right. I think that your first calibration point to decide which option you will use is first to discover what your students believe about the concepts of self-motivation as applied to learning.
As an instructor, I can assign readings to students, but I cannot actually force students to read. I can only provide consequences in the form of good/bad grades by, for instance, testing students over the reading. I believe it would help you to similarly identify which actions you have control over with regards to giving practice exercises.
Students who attempt the exercises before comparing their answer to the solutions could benefit from formative self-assessment, i.e. identifying where they went wrong and how to correct it, or validating correct approaches.
- You could withhold solutions, which would make it difficult for students to self-assess their work, but would also prevent students from only reading the solutions without attempting the exercises.
- You can provide the solutions as well as instructions on how students should use them for the best learning. You cannot force students to follow those instructions, but the reward of good grades (assuming your exam assesses the desired learning outcomes) is a motivation you can provide.