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I want to know if there is any research on whether students' learn better by answering selected response (multiple choice) questions than constructed response (written) questions.

My gut feeling is no. Well-structured written questions bring out much more reflection and learning than multiple choice. There are written questions that cannot be converted to decent multiple choice questions, such as proofs.

However, I may be wrong. Is there any evidence on this?

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    $\begingroup$ I think the terms you are looking for are "constructed response" vs. "selected response". $\endgroup$ – mweiss Jul 29 '16 at 17:35
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    $\begingroup$ I think that the change to the title is actively harmful to reader understanding. Everyone understands what "multiple-choice" is. The opposite of that is probably best called "short answer" or maybe "open response". E.g., "Problem-solving and higher-order reasoning skills are better assessed through short-answer and essay tests." en.wikipedia.org/wiki/Multiple_choice#Disadvantages $\endgroup$ – Daniel R. Collins Jul 29 '16 at 22:17
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    $\begingroup$ Another example, from instructions to NYS Common Core math testing, grades 6-8: "For all three grades, the tests consist of multiple-choice and short- (2-credit) and extended- (3-credit) response questions." -- p12.nysed.gov/assessment/sam/ei/td-68math16.pdf $\endgroup$ – Daniel R. Collins Jul 29 '16 at 22:21
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    $\begingroup$ I'd suggest that SOME multiple choice questions can be written in a way that assesses understanding and successfully exposes common misconceptions, for instance, in the AP curriculum. I have students who can do all the procedural questions get completely tripped up on questions that ask them to pick between five similar, but not identical answers with subtle but important differences. $\endgroup$ – Opal E Jul 30 '16 at 0:00
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    $\begingroup$ I like the question better now. However, you ask about how much students learn, but as if they will only see one of the two. Do you mean to ask about the effect of the assessment style on learning? I would say teaching benefits from both types in different situations. As well as @OpalE 's point about checking subtle points at the end of teaching a subject, multiple choice is good when a topic is first introduced to give the students the confidence to have a go at all. ... $\endgroup$ – Jessica B Jul 30 '16 at 7:40
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In most fields, I think the research indicates the opposite, though I don't know of any research that is directly on point.

One of the weaknesses of multiple choice tests is that they allow the student to write the answer when they recognize it among the choices. This is easier than retrieving it without answer cues, and so the well-established benefits of retrieval practice (a.k.a. the testing effect) for retaining information aren't gained. However, in math, even multiple choice questions typically require some calculation rather than mere recognition, so this is likely less of an issue in math.

While I don't know of research directly on point here either, I also suspect that multiple choice tests give students an excuse to guess on a problem and stop thinking earlier than they might if they had to come up with something rather than leave a free response question blank. Since even unsuccessful initial struggles with a problem promote retention (assuming appropriate feedback is soon provided), I would imagine that this would be a problem even in math.

Also, while very skilled test writers can find ways to write test questions to test a wide range of thinking processes in math, it isn't easy. Even some government or commercial sources often produce multiple choice questions in math whose intent can be evaded by a clever student. (A blatant example: figuring out which choice is the solution to an equation by plugging each choice in turn into the equation rather than solving the equation.)

As for anything that indicates superiority of multiple choice questions in some respect, I don't know of any. They're really used more for convenience than because they're better at promoting learning.

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