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I've been tutoring mathematics at university level for over 10 years, and one of the more common requests from students is worked solutions for sheets of exercises. Most educators I've worked with seem to think that not including worked solutions, and instead encouraging students to ask tutors questions, is the superior method for educating mathematics students. I also subscribe to this belief. I was wondering if there are any studies that either confirm or refute this belief.

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    $\begingroup$ My understanding, and experience, is that students benefit tremendously from worked examples when they are first learning new concepts. There is a nice small text from Michael Pershan called "Teaching Math with Examples" which touches on some of the research and strategies. He cites "Learning from examples: Instructional principles from the worked examples research" (2000) by Atkinson et al, "Learning how to solve problems by studying examples" (2019) by Gog et al, and others. $\endgroup$
    – Carser
    Aug 23 '21 at 19:08
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    $\begingroup$ A PDF of the former can be found here. Reading through the conclusion, this seems to partially answer my question, as the survey cites studies that show that worked problems can indeed foster adaptive learning. However (and some deeper reading will be required), I haven't yet found a comparison between having worked solutions available and having tutors available for one-on-one tuition. Thank you for the references, @Carser! $\endgroup$ Aug 23 '21 at 19:22
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    $\begingroup$ Regardless of what studies may show, they would only indicate general average tendencies for larger populations of students (i.e. an average over many classes). My personal experience is that for some classes (the specific topic being taught and the specific group of students are the variables) the absence of solutions is best, and for other classes the presence of solutions is best (although in these cases, "bald answers" without all the details worked out is usually best). The hard part is deciding for a particular class which of the two is better. $\endgroup$ Aug 23 '21 at 21:16
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    $\begingroup$ In my experience, "worked solutions" is insufficient. What is most useful is solutions with accompanying explanations from a teacher with experience of the common pitfalls. $\endgroup$ Aug 23 '21 at 21:48
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    $\begingroup$ I guess that the usefulness of "worked solutions" depends on many factors. (1) Are the exercises (proof tasks or computations) similar to what the students have already seen during the lecture? If yes, then worked solutions are "more of the same", if not (i.e., if the lecture concentrates on deriving results whereas the exercises concentrate on applying them), worked solutions are probably more important. ... $\endgroup$
    – Uwe
    Aug 25 '21 at 8:17

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