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My mathematics education was essentially rote, you learned the formulas and applied them almost algorithmically to the problems you were presented with; the teacher dictated a method and you followed it. This conveyed a dry and static picture of mathematics with no room for creativity and where curiosity was strait-jacketed and questions weren't encouraged.

I feel when I approach problems now the seeds of this have come to bear. I think very linearly; I see a certain method I can apply to a problem and I keep attacking the problem with the same method till either I give up or the problem beats me. It is almost like tunnel vision, I see things very narrowly. The saying everything looks like a nail when you have a hammer comes to mind. I feel this is particularly prevalent when I'm tackling Olympiad problems, I feel certain students have a fount of creativity they can dip into, mines in contrast seems depleted.

I assume a good pedagogy would engender a broad, deep and intuitive understanding of mathematics and would enable you to think more creatively and draw on broader areas of mathematics when attempting to tackle a problem and would further allow you to think spontaneously when something out of the norm is encountered.


How does one concretely cultivate this? In essence what I am asking is:

  1. How does one think creatively when it comes to problem solving in mathematics? Can I teach myself to think creatively?

  2. Are there any books or resources that are particularly helpful with regards to this?

  3. What are the pre-requisites to creative thinking? Does one have to be deeply proficient in a subject and have a good grasp of its concept before they can be creative/inventive? Is the creativity I perceive in others just subject proficiency?

  4. Or does one just have to be "efficient" (as opposed to proficient) to be creative?

  5. Can creativity be learnt? Or are people inherently creative: either you have it or don't?

  6. Are there any ways to correct the tunnel vision I described, that will enable me to think more open and broadly and just generally shake off the ill effects of a poor education? (I guess this is what I attribute my inadequacies to).

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  • $\begingroup$ I think it's better to let them do creative thinking. Sometimes, when they're directly told how to show the work on something or a long list of instructions for calculating the sum of fractions, they are confused and struggle. At first, we could just teach them natural numbers and focus on natural numbers for a really long time. Then they can be introduced to the numbers that can be gotten through repeated division by 2. Then the rest of the real numbers can be constructed. We should use Polish notation with no spaces. ⌀ should represent the number 0. Each digit from 0 to 7 should represent a $\endgroup$ – Timothy Mar 11 at 19:14
  • $\begingroup$ unary operation of first left multiplying by 8 then left adding a number from 1 to 7. In my version of Polish notation, a left subtraction is an inverse of a left addition and a left division is an inverse of a left multiplication. Then they could be left to teach themselves how to add numbers given in rational notation where the rational notation of a number is an expression of the form of division and each operand is an integer expressed as a string of digits followed by ⌀, called the octal notation. So they could teach themselves that +÷4⌀3⌀÷6⌀5⌀ = +÷41⌀11⌀÷41⌀21⌀ = ÷41⌀+11⌀21⌀ = ÷41⌀32⌀. $\endgroup$ – Timothy Mar 11 at 19:23
  • $\begingroup$ They should be given very little homework so that they can take the time to think bout each homework problem slowly and carefully. Later, they could be given a homework problem to teach themselves how to calculate what ÷÷2⌀3⌀÷5⌀7⌀ is in rational notation. Then they will figure out that the answer is ÷71⌀61⌀. That they should be able to figure out without too much trouble. However, if they're pushed to learn how to show the work and explain things, they might struggle and learn even less because teaching is not learning. Finland has the best education system. Maybe you could discuss these ideas $\endgroup$ – Timothy Mar 11 at 19:29
  • $\begingroup$ with the Finnish national agency for education. Also, we could have a competitive market for all sorts of jobs including education except that the most successful companies with the highest reputation will be the ones that do jobs for free end terminate employees whose total income other than having certain types of costs covered exceeds a certain amount and doesn't compete to be the best and will work with any company that doesn't have a flaw in the way they work. The public will trust the companies that have a maximum income policy more. The public probably doesn't trust for profit $\endgroup$ – Timothy Mar 11 at 19:33
  • $\begingroup$ companies to do a very good job of acting on the climate crisis. Some companies may actually have good ideas. It will be to their own personal benefit to do a better job of solving the climate crisis by suggesting a change in law where any company may irreversibly surrender the ability for any of their employees to exceed the maximum income from now on without being terminated from employment and then each company will then have an incentive to surrender the ability to exceed the maximum income so that their ideas will be trusted more and listened to more and since they can no longer get such $\endgroup$ – Timothy Mar 11 at 19:37
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The question is very broad, as you surely realize, since you broke it up into six sub-questions (which are also very broad!). In order to attempt an answer, I will have to send you on a bit of a scavenger hunt:

First, see my answer here and note that I have a literature review from my doctoral dissertation, which was entitled Conceptions of Creativity in Elementary School Mathematical Problem Posing. Practice with formulating your own problems is one way in which you can foster creativity; the literature review contains many references you can follow up on with regard to creativity, mathematical problem posing, or both. Moreover, you can find further remarks of mine about problem posing here.

Another potential way to cultivate creativity is to impose constraints on your work; to this end, check PD Stokes' (non-mathematical) book Creativity from Constraints and see if you find it helpful.

Other good sources are some of the work by B Sriraman and his collaborators; most recently, I received a recommendation for a book by P Borwein et al that I have yet to check out, Mathematicians on Creativity, which is described as follows:

This book aims to shine a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-caliber working mathematicians. In their own words, they discuss the art and practice of their work. This approach highlights creative components of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work.

If you like creative people's recollections, then a non-mathematical source (though it includes a nice piece by Feynman) is F Barron's Creators on Creating. Of course, one cannot always take self-reported stories at face value; a discussion of this matter and how storytelling relates to creativity can be found in work by MC Bateson.

More generally, if you are going to go with a single textbook to gain a broad sense about creativity and the different ways in which it is conceived of (especially as related to problem solving) see Weisberg's text mentioned in my answer here. Again, this is not explicitly about mathematics - in fact, there is a dearth of work on STEM creativity! - but you may well find tidibits of interest. (And there are at least some mentions of the hard sciences, e.g., genetics, biology.)

As far as whether people are inherently creative: No, I do not believe so, though any question about the construct will likely depend on the definition (or conception) you have in mind. One creativity theorist of whom I am particularly fond is Howard Gruber; he effectively defines creativity as purposeful work, and this is not something we are born able to churn out. Purpose is something that is developed over time, and Gruber works to "demystify creativity" as many others who use case studies are inclined towards. For a couple more of my answers that mention Gruber and his work, see here (less relevant) or here.

Though following the links above (and links therein!) will already entail quite a bit of reading, I thought I would voice one pet peeve of mine: Many studies in the mainstream media that are purportedly about creativity use the Torrance Tests of Creative Thinking (TTCT) as their main instrument. However, the TTCT really measures what is today known as divergent thinking (DT) and, for those who research creativity, this is no longer thought to be the same as creativity. In fact, such an identification should have been put to rest in the 1970s (!).

Even (or especially) major researchers of DT - such as Runco - will tell you this. For example, see (or at least glean the essence from the title) his Commentary: Divergent Thinking Is Not Synonymous With Creativity (pdf). One must be wary of studies that use TTCT-like assessments to measure DT, and then form strong (i.e., unfounded) conclusions about creativity.

If you are interested in other ways to assess creativity: I am somewhat fond of TM Amabile's Consensual Assessment Technique (CAT) for which you can find a strong overview article by Baer and McKool here (pdf) that has a fair bit in the context of higher education. Insofar as the above-mentioned TTCT is concerned, note that the authors write in their introduction:

If creativity is to be assessed in college settings in a meaningful way, divergent-thinking tests like the Torrance Tests of Creative Thinking and other commonly used creativity tests are inadequate because they fail to meet even the loosest standards of validity.

Not all of what is written in the mainstream media about creativity is as unpalatable as some of the TTCT based work. A recent piece by Brooks in The New York Times called The Creative Climate includes the following interesting snippet:

Today we live in a distinct sort of creative environment. People don’t so much live in the contradiction between competing worldviews. We live in a period of disillusion and distrust of institutions.

This has created two reactions. Some monads withdraw back into the purity of their own subcultures. But others push themselves into the rotting institutions they want to reinvent. If you are looking for people who are going to be creative in the current climate, I’d look for people who are disillusioned with politics even as they go into it; who are disenchanted with contemporary worship, even as they join the church; who are disgusted by finance even as they work in finance. These people believe in the goals of their systems but detest how they function. They contain the anxious contradictions between disillusionment and hope.

I find the above excerpt quite interesting for a few reasons. One, it relates to some of the other literature on creativity: the anxious confrontation alluded to in the last sentence reminds me of R May's work in existential psychology, and especially The Courage to Create (pdf). (The word monads also comes up in Darwin's work; cf. Gruber's book Darwin on Man.) Two, the notion of distrusting institutions is an important theme in Latour's An Inquiry into Modes of Existence, which is part of an on-going collaborative project of potential interest to those who study creativity. Its corresponding site, for which registration is free, contains the entire book; it also contains a video of a mathematician (darij grinberg) writing a question and uploading it to MathOverflow (the specific question can be seen here).

Perhaps it would be best to curtail myself here; my apologies for the lengthy nature of this response.

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    $\begingroup$ Thank you for your detailed answer, I'll read it when I'm free, thank you! $\endgroup$ – seeker Jul 30 '14 at 12:05
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There are studies indicating that metacognition training can improve problem attack. In the linked article by Schoenfield (which I may have gotten from here, I'm not sure), he discusses the differences in how experienced mathematicians approach a problem compared to inexperienced students, characterizing each approach by the stages "Read, Analyze, Explore, Plan, Implement, Verify." An inexperienced problem-solver will stay in the "Explore" stage, but with some training, can be taught to use planning, implementation, and verification to improve their problem solving skills.

Another process which is highly cited is George Polya's Problem Solving Process where he describes four stages he claims form the problem-solving process: Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back. While clearly a bit behind where you are since you are doing Olympiad-level mathematics, it may come in handy to someone else looking for problem-solving strategies.

I also personally think that metacognitive skills can vastly improve problem-solving performance, but also exposure to similar problems can as well. Sometimes there is a "trick" that you wouldn't normally think of except you saw it used once in a similar problem. The evaluation of the integral $\int_0^{\infty} e^{-x^2}$ comes to mind. I also recall some problems on the Putnam which would have been infinitely easier had a particular result from, say, graph theory come to mind (For the record, I have little experience in competitive mathematics and my participation was purely for fun). In that sense, familiarity with background information, too, is important. It would take someone with considerable mathematical prowess to be able to derive new theorems in time to complete a competitive mathematics exam.

The three factors I have mentioned probably only scratch the surface, but all three are things that can be worked on and improved and do not necessarily require innate ability.

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If you are looking for a book you can use to study an effective practice of teaching mathematics in a way that is definitely not intended to leave students with mathematical tunnel vision, can I recommend Teaching Problems and the Problems of Teaching by Magdalene Lampert?

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    $\begingroup$ Why is this a relevant recommendation? i.e., How does this book specifically help to address the OP's questions? $\endgroup$ – Benjamin Dickman Aug 14 '14 at 11:15
  • $\begingroup$ If you teach through problem solving, that engenders an entirely different mindset than what you get by teaching 'content'. I haven't read this book, but it does sound like it would address the question - at a very different level than your answers. $\endgroup$ – Sue VanHattum Sep 5 '14 at 14:58
  • $\begingroup$ Yes, this is my point. This book describes a way of teaching mathematics which is decidedly not linear and not intended to result in tunnel vision by students. An aside: It is my understanding that the more you know, and the more connections you are able to make between what you know, the more creative you are. EG. A different technique for ensuring that students understand more mathematics will result in more creative uses of mathematics, although one should probably provide time to ensure that students have time to apply their knowledge creatively. $\endgroup$ – David Wees Sep 6 '14 at 16:08

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