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An old favourite topic of mine to daydream about on pleasant afternoons is this:

If you could completely redesign the university-level mathematics curriculum from the ground up to be as good as it could be, but without external constraints (e.g. designing a math curriculum for an isolated group of mountain monks with no interest in employment or in going to universities, but wouldn't mind winning Fields Medals), how would you do it?

Constraints are as follows:

  1. The students do not necessarily have good pre-college background, and are not necessarily of "high math aptitude", whatever that means. So the curriculum should be easier than a regular undergraduate mathematics curriculum.
  2. There is no need to break things up into courses, and there is no need to grade or to write tests. Thus, nothing ever needs to be taught "because it's easy to write easily-graded exercizes about it".
  3. There is no back-compatibility requirement. No need to conform to anyone's expectations of what student ought to learn or what will land them nice jobs.
  4. Students are assumed to be highly motivated and to always pay attention.
  5. There is no need to restrict to "pure math". A physics or engineering or finance example or technique or insight are entirely admissible.
  6. The objective is to maximize student's understanding of mathematics and their abilities both to construct theories (Grothendieck) and to solve problems (Erdos).

Relaxing some of these constraints, and in particular assuming students of high mathematical aptitude, an attempt along these lines might include Bourbaki's work or Math55 in Harvard, although these are strongly biased in favour of "pure math" and are hard.

I posted on this topic on Facebook, and received a wonderful array of comments from mathematicians and math educator friends. The ideas were interesting, and collecting more would also be fantastic (but that isn't the question). I also saw this intriguing argument by Alexandre Eremenko that vector calculus ought to be dropped in favour of differential forms.

My actual question follows below:

Question: Has there been an attempt to design a utopian mathematics undergraduate-level curriculum, one of whose goals is to be easier for students than current curricula, with no external constraints (in particular, no need to test students and no compatibility constraints)? Has there been any research on the topic of what such a curriculum would include and how specifically it would differ from existing curricula? Could you point me towards such references?
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    $\begingroup$ I would be surprised if such a theoretical curriculum existed. What would it gain besides a what-if scenario? Also, the question might be too opinion based. $\endgroup$ – Chris C Apr 1 '15 at 13:48
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    $\begingroup$ The gain would be as a reference point, to which less-utopian currlicula could turn to for inspiration. I am also not entirely convinced that this utopia cannot exist- I've been toying with the idea of teaching math to certain groups of Haredi (Ultra-Orthodox Jewish) yeshiva students, who would never attend a university in a month of sundays, but highly respect abstract knowledge for its own sake, and for that audience I would expect basically the kind of constraints I listed- so while not being an immediate need, this is also not an entirely theoretical question. $\endgroup$ – Daniel Moskovich Apr 1 '15 at 14:20
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    $\begingroup$ I'm voting to close as too broad. This is an invitation to open-ended discussion and IMO is not a good fit for the stackexchange format. $\endgroup$ – Ben Crowell Apr 1 '15 at 15:25
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    $\begingroup$ I think, as currently stated, the question is too broad, open ended, and opinion based. If you just asked about education research on such a topic (i.e. just any potential reference request), that would be much more appropriate. $\endgroup$ – Chris C Apr 1 '15 at 17:12
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    $\begingroup$ Peter Saveliev's fantasy math musings could be of interest. $\endgroup$ – J W Apr 4 '15 at 8:58
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Years ago, as an undergraduate student, I experienced something close to your description of a utopian mathematics undergraduate-level curriculum at Sharif University of Technology in Iran. We didn't mind winning Fields Medal. Indeed,one did: Maryam Mirzakhani. I remember, one of the courses we had was "geometric analysis", quite uncommon as an undergraduate course.

I believe the reason that such a curriculum was designed was that the department had attracted nearly all Iranian medalists of The International Mathematical Olympiad and also a good numbers of some other highly motivated students. In a sense, as a student, you had a choice, to follow the enhanced curriculum or the standard one. Though in a sense the enhanced curriculum was very successful (apart from Maryam, I can name many other great mathematicians who experienced that programme), finally it ceased! I guess the reason was that the students of the other tribe (those who satisfied your first constraint) outnumbered and also running two parallel curriculum had many social issues.

Added. As I said in my comment below your question inspired me to document the history of those old good days. In that direction, I made a wiki in which people can write about what they experienced. It has just been started and most of the contributions at the moment has been naturally written in Persian. However, there are a few in English that I am sure give you some ideas of how strange was those days. Here is the link to one of them: Why some of us became number theorists

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  • $\begingroup$ Do you have a link to what the content of the enhanced curriculum was, and how it differed from the "standard curriculum"? $\endgroup$ – Daniel Moskovich Apr 2 '15 at 11:09
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    $\begingroup$ Indeed, your question inspired me to find a way to document the history of those "golden" years. Unfortunately, we are not good at documenting things. They are coming and going and after a while no one remember what they were and why they happened. I am going to contact with those who experienced that "enhanced curriculum" and gather whatever I can. I'll let you know as soon as I can. $\endgroup$ – Amir Asghari Apr 2 '15 at 11:32
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    $\begingroup$ One main difference between the standard one and the enhanced one was in the way you could take the courses. To take a course in the standard route it was required to pass all the prerequisites. For example, to take Analysis 1, you should have passed Cal 1, 2, and Foundation of Mathematics. But, in "enhanced curriculum" you could take "geometric analysis" or any master of PhD courses you wished, providing that you and your instructor believed you could. $\endgroup$ – Amir Asghari Apr 2 '15 at 11:41
  • $\begingroup$ @AmirAsghari, but the selection of courses (and presumably the instructors, and perhaps even more important, the classmates) were certainly extraordinary. $\endgroup$ – vonbrand Dec 30 '15 at 12:01
  • $\begingroup$ @vonbrand you are absolutely correct about classmates. But, the rest wasn't that much systematic and more relied on a few extraordinary instructors who were planning the courses, defining the syllabuses and lecturing mathematics with joy. $\endgroup$ – Amir Asghari Dec 30 '15 at 14:46
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From my dim and tattered memory pops out the name of Stephen Leacock. Assuming I got it right, he had some similar ideas placed on paper. The economy of the situation reduced several departments and personnel to a large sitting room with comfortable chairs, a ready supply of cigars, ashtrays, and perhaps libation, and classes and lectures converted to discussions and reflections (sort of like the old Socratic school, but with better furniture).

In spite of the satirical nature of his essay (which I hope someone will recall), I think he anticipated many of your points, and that in this modern day and age, a utopian university would be electronic, hook up a student with a variety of materials as well as an appropriate set of tutors on a variety of subjects, and make the university a very large virtual sitting room, much as What's-His-Face wrote about many years ago.

Gerhard "Typed With A Straight Face" Paseman, 2015.04.01

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    $\begingroup$ My memory is poorer than I thought. I won't swear to it, but I think the passage I can't quite recall comes from Oxford As I See It (readbookonline.net/readOnLine/63418 ), near the bottom, beginning with "If I were founding a university...". I have discovered some other essays, e.g. Education Made Agreeable...(readbookonline.net/readOnLine/63447 ), which I think should be required reading for mathematics educators. Gerhard "April Fool's Jokes Are Serious" Paseman, 2015.04.02 $\endgroup$ – Gerhard Paseman Apr 2 '15 at 17:49
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While not entirely free of "external constraints", for many years now, I have been working on a three semester sequence [5-4-4] intended to parallel the Arithmetic - Basic Algebra - Intermediate Algebra - Precalculus 1 - Precalculus 2 - Calculus 1 sequence. The idea is to lead into the integrated sequence I mentioned elsewhere on this site (I don't know how to link to it) by integrating Arithmetic - Basic Algebra - Intermediate Algebra into a conceptual flow also centered on approximations.

I wanted the "external constraints" as a reality check. That is, I extracted, and teach every semester, two "standalones" from the sequence to see how students take it.

See the Notices for a relevant article and, for the materials themselves, FreeMathTexts.org.

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I can't speak to existing research. But, I can't help but think of Frenkel's account of the community of mathematicians in the time of the cold war. The idea of meeting at night with a large group of practicing mathematicians to engage in mathematical conversation, guided by the masters, it's fascinating. Perhaps this still exists at some elite universities, but, in my limited experience the closest thing is a department colloquium. I suppose in graduate school, there is a time before the thesis needs to be written and jobs need to be found, when not studying for quals, where we just brainstorm with our peers. But, it seems to me, the time and social aspect of the Russian school is much more serious. It seems math was their priority.

To me, from a math perspective, that is utopia. In other words, the MSE and MO are a bit of utopia for us to enjoy when the tasks of life allow us leave to play.

The other aspect of Frenkel's account which merits our attention is the mentor he acquired early in his highschool (I forget the precise details, and one of my students currently has my copy of Love and Math which you should read if you haven't already). This mentor skipped past all the usual topics and immediately challenged him with interesting, cutting edge, math. Now, obviously the mentor had to know that Frenkel had unusual ability. It's interesting to read about how the mentor enticed Frenkel deeper and deeper into the world of beautiful mathematics.

Let me collect some guidelines which are useful.

  1. Find some community of mathematics where you can sharpen your math by the constructive criticism of others. Math is a social enterprise at a certain level, it is nice to have some sense of what other like-minded individuals find worthwhile. Here I mostly mean for the student to develop a mature style, their ideas must ultimately be their own. A good REU experience, or a cluster of excellent students can go a way towards the goal I outline here. It's equally important to learn how not to be a jerk, and, yet, to still say something.
  2. Find a master. Learn everything they know. Repeat.
  3. Free your mind of this nonsense like "I'm an applied mathematician" or "I like algebra, but not analysis" or (my vice early on) "I'm just interested in calculations, tell me how to calculate". Embrace a liberal arts education in mathematics. Learn everything.

You might criticize my answer in that it is not about a community, it's really just advice for an individual. But, that is my last and most important point: one size does not fit all. Education, especially utopian, must be individualized.

This is why homeschooling, when done correctly, will dominate conventional schooling. Especially now that we have shackled teachers from enforcing difficult standards...but, I probably digress.

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