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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.

Theo Bendit
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  • This is my first question on MESE. Please let me know if there's any way I should improve my question. – Theo Bendit Aug 23 '21 at 18:52
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    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. – Carser Aug 23 '21 at 19:08
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    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! – Theo Bendit Aug 23 '21 at 19:22
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    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. – Dave L Renfro Aug 23 '21 at 21:16
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    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. – Joseph O'Rourke Aug 23 '21 at 21:48
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    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. ... – Uwe Aug 25 '21 at 08:17
  • (2) Do students who are not able to solve an exercise see the full solution in a tutorial, do they get a printed copy, or do they only get hints, even if they are not able to solve the problem with these hints? The first choice is probably better than the second one (see Joseph O'Rourke's comment), but the second one may still be better than the third one. ... – Uwe Aug 25 '21 at 08:17
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    (3) Are the exam questions very similar to the exercises or not? In other words, do the exams test whether students know the solutions to the exercises, or whether they are able to find solutions themselves? In the first case, worked solutions are a useful model; in the second case, students have to learn to solve problems themselves, and the exercises should give them the opportunity to learn that. – Uwe Aug 25 '21 at 08:17
  • When having a question like that, just try both ways: provide solutions to some types of exercises but not to other ones (let a random number generator choose the types, so that the experiment will not reflect your personal biases). Then in a couple of months look at whether the students make more progress with one type than with the other, and if you see some noticeable difference, just accept the practice that works better whatever it is. All educational studies should be viewed as "ideas for change/improvement" but whether they work or not is best determined by trial and error. – fedja Oct 31 '22 at 01:27

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