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In Jennifer Roberts' article The Power of Patience: Teaching students the value of deceleration and immersive attention she talks about intentionally slowing down to contemplate deeply a work of art. To my mind, the study of mathematics also benefits from slow and careful thought in a distraction-free environment, in addition to spending time on interactive discussion and group work.

Has research been carried out on immersive attention or sustained concentration when learning mathematics? If so, what were the results and are there any useful suggestions for teachers and students, both inside and outside of the classroom? This could include lesson pacing, variation of activities and selection of assignments, for instance.

Also, does anyone have experiences to share of the process and results of setting students an exercise that involved contemplation of a mathematical object or theorem for a seemingly-excessive length of time a la Roberts' example in the above-mentioned article?

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  • $\begingroup$ Is "immersive attention" something about the environment (i.e. freedom from distractions) or something about one's orientation toward mathematics (persistence in problem solving)? Or do you mean making problems about more than finding answers, but rather contemplating the problem beyond a solution to a particular instance of the problem? $\endgroup$ – JPBurke Oct 25 '14 at 16:30
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    $\begingroup$ I think all three are involved, but perhaps especially contemplation to aid in solving difficult problems and in generalizing. It's also about the opposite of working under time pressure, such as timed drills. $\endgroup$ – J W Oct 25 '14 at 18:01
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    $\begingroup$ I found the article flawed in its elitist attitude to maths education. I suspect a think-tank of honours pure maths students would enjoy deep immersion as this is their thought style. For the rest of the world maths remains elitist, boring, and out of reach. Making it fun lasts till about grade 6 and then teen cynicism starts to bite. When maths classes are stratified the lower classes are universally perceived as "dumb", an attitude reinforced by teachers who in the main crave academic relevance. $\endgroup$ – Peter Russell Dec 30 '14 at 13:18
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    $\begingroup$ @PeterRussell I'm somewhat puzzled by your opening sentence "I found the article flawed in its elitist attitude to maths education", as Roberts' writes from an art history perspective and does not mention mathematics education. $\endgroup$ – J W Dec 30 '14 at 13:33
  • $\begingroup$ This is the second paragraph of @PeterRussell's answer that I convert to a comment: "I don't know that maths can be taught to all, and maybe in a civilised society we should be better at recognising the intate skills and strengths of students and stop procrastinating the academic standards of last century as if it was, or ever will be available to all, if only they buckled down and tried a little harder." $\endgroup$ – quid Dec 30 '14 at 19:26
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Malcolm Gladwell addresses this issue briefly in Outliers. Here I take some quotes from a Zook Tutoring posting, quoting from Outliers:

[...] Alan Schoenfeld, a math professor at Berkeley, made a videotape of a woman named Renee as she was trying to solve a math problem. [...] Twenty-two minutes pass from the moment Renee begins playing with the computer program to the moment she says, “Ahhhh. That means something now.” [...]

We sometimes think of being good at mathematics as an innate ability. You either have “it” or you don’t. But to Schoenfeld, it’s not so much ability as attitude. You master mathematics if you’re willing to try. That’s what Schoenfeld attempts to teach his students. Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds.

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Since posting my question, I have come across some material on contemplative mathematics that I will share as an admittedly-incomplete answer:

In Section 3 of On Contemplation in Mathematics (Wolcott, 2013), Luke Wolcott discusses meditation in education and describes an informal experiment he conducted in an introductory differential equations class, as well as the feedback collected on a weekly basis. See also Mindfulness at Lawrence with Wolcott and Justin Brody's Contemplating Infinity. On the latter topic, I was intrigued to find out about Gardens of Infinity, a collaboration between Wolcott and interaction designer Luke Dressel.

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  • $\begingroup$ "On Contemplation in Mathematics" seems interesting. Given the success that mindfulness techniques have had in recent years treating psychological problems such as borderline personality disorder, it isn't too off-the-wall to think that some variation of it can help with math phobia, at least for some students. $\endgroup$ – John Coleman Aug 10 '16 at 15:33
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Learning anything needs patience and continuous effort.

If we talk about Mathematics, It is having Abstract things. We accept one as symbol "1" in it. To teach mathematics most important thing is you have to mix yourself with nature. There are patterns and shapes in nature which will help you to understand and taught others. For example, shape of archimedian spiral originally it is derived from the shell of a nautilus.

Check this history of mathematics on youtube

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    $\begingroup$ Could you explain briefly what is to be found if one follows that link? People should know where an external link will take them, and the explanation will remain helpful even if the link dies. $\endgroup$ – Joonas Ilmavirta Dec 18 '15 at 10:50
  • $\begingroup$ @JoonasIlmavirta thank you for suggestion brother. $\endgroup$ – Divyek Dec 18 '15 at 12:09
  • $\begingroup$ I agree with the patience and continuous effort, but how the rest of your answer related to the question? Besides, there are a number of concepts and procedures in math that you cannot "mix yourself with nature". $\endgroup$ – Mark Fantini Aug 11 '16 at 2:04

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