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I came to mathematics via physics, in part because of the reputation of physics as allowing "non-rigorous" reasoning. The subject felt more free and less anal-retentive than mathematics. This is not to say that physics does not require care, but in many physics books, proofs are scarce and derivations plentiful. Arguments are given only so far as needed to convince the student of the truth of something under the "nicest" possible conditions. When I was young, this was appealing. The sense that I only needed to argue to a point where I was convinced led me to revisit concepts and computations far more often than if some taskmaster were standing over me with a ruler waiting for me to make an error I wasn't prepared (read: mature enough) to acknowledge.

Of course, now I am a mathematics professor and I take a far different approach with my students. I require them to understand things that (based on their performance on assignments) they really do not appreciate to a degree I find satisfactory. Their basic skills are often abysmal, and the impression I get is that, if they were not forced to, they would never revisit anything mathematical. Many students see no beauty in this subject, only fear and a need to be right and get "the answer".

This condition, and its contrast with my own rather lax beginnings in the subject, makes me wonder whether we should be doing something vastly different than we are doing.

Question: Has an approach to mathematics education been tried that explicitly deemphasizes correctness and assesses almost solely based on exploration, interest and beauty? If so, how does one implement such an approach?

Of course, in the end, we want students to appreciate the need to be careful and precise. My implicit claim is that by providing an enjoyable experience for students, they will voluntarily think about topics again and again and therefore will "converge" to correctness via willing and frequent re-exposure. Is this general view valid in any sense? What research supports or critiques this idea?

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    $\begingroup$ I might politely disagree with the dichotomy expressed here. You can have conceptual rigor without needing to agree that math is essentially "beautiful" or "enjoyable". $\endgroup$ – Daniel R. Collins Dec 18 '16 at 5:02
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    $\begingroup$ I think this question is overly broad. Primary, secondary, early undergraduate and late undergraduate education are extremely different from one another with respect to their approach to rigor and "correctness". Exploration-based approaches are much more common in the earlier grades. $\endgroup$ – mweiss Dec 18 '16 at 15:33
  • $\begingroup$ @mweiss: This is good to know. $\endgroup$ – Jon Bannon Dec 18 '16 at 18:35
  • $\begingroup$ @DanielR.Collins: You are perhaps right. I'm not interested in that kind of math, though. $\endgroup$ – Jon Bannon Dec 18 '16 at 18:35
  • $\begingroup$ I should probably edit the question. (I posted it after grading a freshman final...brrrr!) What I am going for is this: How can we assess a course so that students will WANT to be precise because of what the concepts dictate...rather than just getting the answer? IBL is the best I've found, so far, for making this shift. $\endgroup$ – Jon Bannon Dec 18 '16 at 18:37
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Here is my attempt to answer your question, if not directly, then at least in spirit! I am specifically responding to the following:

Many students see no beauty in this subject, only fear and a need to be right and get "the answer".

Question: Has an approach to mathematics education been tried that explicitly deemphasizes correctness ... ?

What research supports or critiques this idea?

Something that I have done to move away from this need to get the right answer is, rather than directly "deemphasizing correctness," trying instead to emphasize the importance of mistake-making in learning (mathematics). For example, I introduce the following norm/expectation on Day 1 of my classes (and have it written out on the corresponding class sites):

Mistake-Making

I expect that we will all make mistakes as we wrestle with new (and sometimes old) ideas, and will use these mistakes as opportunities for learning. There is a large body of research on mathematics, and other subjects, that says we learn through mistakes — after all, when your work is mistake-free it only demonstrates what you already know!

I expect we will take intellectual risks and make mistakes in the process, and that we will do our best to embrace the ability to honor, even celebrate, being wrong.

For research to support this idea, I would point you to Jo Boaler (for mathematics and mathematical mindset, in particular) or Carol Dweck (for mindset research, more generally). Boaler has, for example, a mention on this Mistakes Grow Your Brain page of the following paper:

Moser, J. S., Schroder, H. S., Heeter, C., Moran, T. P., & Lee, Y. H. (2011). Mind Your Errors Evidence for a Neural Mechanism Linking Growth Mind-Set to Adaptive Posterror Adjustments. Psychological Science. Link.

and a mathematics-specific reference to:

Steuer, G., Rosentritt-Brunn, G., & Dresel, M. (2013). Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate. Contemporary Educational Psychology, 38(3), 196-210.

The latter is summarized by Boaler as:

"Gabriele Steuer and her colleagues looked at the climate of math classrooms to consider the impact of “mistakes friendly” or “mistakes unfriendly” environments on students’ reactions to errors and the amount of effort they would put into classes (Steur et al., 2013). They found that when students perceived their classroom as mistakes friendly – above and beyond other aspects of their classrooms environment – they increased their effort in their work."

Anecdotally, I certainly observe an increased willingness by students to share their work and not obsess over the right answer. When I ask for volunteers to discuss some idea they have just had, or problem they have just attempted, I often hear a comment to the effect of, "I'll come up to the board! I think a made mistake, though," to which I reply, "Great!" Indeed, sometimes there is a mistake - but then others can chime in. In fact, sometimes a student realizes the way she began the problem was erroneous, and - rather than saying "whoops, never mind!" - will end up solving it out on the board for the first time (or, at least, re-attempting it).

I should also note that, personally, I find the philosophy of mistake-making as a plus to be anxiety reducing. Neither students nor teachers need to obsess over correctness; the notion that getting something wrong can be helpful reduces the burden of initial precision. Still, I caveat that, one, instructors should perhaps be careful about taking this to an extreme (e.g., being unprepared as a teacher and having one's work constantly full of careless errors), and, two, this is not a way to "deemphasize correctness" permanently - rather, it is the notion that errors play a natural, and helpful, role in facilitating students' convergence to correctness (to pilfer wording from the OP) as they wrestle with new mathematics and mathematical ideas.

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    $\begingroup$ I agree wholeheartedly about mistake-making, which I've observed in my own learning and in my students'. The best students often make mistakes: they make them early and they learn from them. I'm not sure whether this approach emphasizes or de-emphasizes correctness, though. My feeling is that correctness is and should be the goal, and this should be valued early on. Without correctness, reflection, consolidation, generalization and such seem difficult. For me, your description of mistake-making puts it in a good light, as something to seek after, and the path to it is built on mistakes. $\endgroup$ – user1527 Dec 26 '16 at 0:43
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    $\begingroup$ @Benjamin: This is great. I agree with Michael E2 that it is probably the emphasis on discovery process/finding correctness/making mistakes that is what I actually wanted... $\endgroup$ – Jon Bannon Dec 26 '16 at 12:33
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I'm sure I've posted this elsewhere, but (in somewhat different ways) two curricula that try for this, I think, are

Note that there are probably lots of others (and I hope other posters mention them), and also note that I'm not sure that they are an approach to the entirety of undergraduate mathematics education. Most "inquiry-based" methodologies for the entire curriculum end up emphasizing correctness and beauty. (Which I like. But that wasn't what you wanted, I think.)

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  • $\begingroup$ @krisman: I personally am not capable of doing what I asked (correctness is essential, in my grown-up opinion, to even see the beauty in mathematics). That said, I remember a time when it was not. What I am asking about is an issue of what one of my wise professors called "truth in teaching". Arguably, most calculus sequences (at least Harvard calculus) try to do what I am asking...in that the concept of limit we allow students to run with is somewhat ridiculous, but "intuitive". This doesn't satisfy me, though. I should ask: How do we get students to want the right kinds of things right? $\endgroup$ – Jon Bannon Dec 18 '16 at 18:43
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    $\begingroup$ Well, Harvard calculus is definitely not "typical" (if you mean Hughes-Hallett et al.); I think most places still use Stewart or some variant thereof. I don't think you can "get students to want", you can just bring them further along that line. But now we get to experiences difficult to convey in such a limited medium as ME.SE :) $\endgroup$ – kcrisman Dec 19 '16 at 13:50
  • $\begingroup$ I do mean Hughes-Hallett et. al. $\endgroup$ – Jon Bannon Dec 19 '16 at 20:13

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