# How can mathematics educators encourage innovation and creativity?

Almost by definition, innovation requires that things be done differently than established custom has it, and comes from the young more often than from the old. In a field as old and established as mathematics, it can be difficult to find something genuinely new that hasn't been discovered, and it may require fairly advanced education to get to the forefront of an area were new mathematics is being created and studied.

However, the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way, and many potentially creative students are likely to be stifled and discouraged long before getting a PhD and getting published.

So, how can educators encourage creativity without sacrificing correctness?

• First and foremost, by being innovative and creative themselves! – dtldarek Mar 21 '14 at 19:38
• I think this is a great question, and wish I had a good answer. I have ideas about how to minimize the "math is just about getting the right answer" component, but not necessarily about effectivrly promoting creativity. Would that still be appropriate as an answer here? – Brendan W. Sullivan Mar 22 '14 at 1:49
• Fine with me, until someone comes up with a better one. This is an open-ended type of question. – Confutus Mar 22 '14 at 2:09
• research/theory eg MathOverflow & Theoretical Computer Science – vzn Mar 29 '14 at 20:21
• Wish I knew. I try to use HW problems that have, in addition to the solution you get by following a given algorithm, also short cuts. Then I spend time in HW session polling students for ideas of alternative solutions. Sometimes it works, but not always. Also one has to be careful not to overdo it lest students get the impression that there will always be a shortcut. – Jyrki Lahtonen May 25 '14 at 5:54

Edit (5/24/14): For the reader interested in a somewhat longer answer, I am including the literature review (and all references) from my thesis on conceptions of creativity with regard to problems posed from the multiplication table. A copy of the excerpt can be found here. My original, shorter answer remains un-edited below.

The story of mathematical creativity is long and probably deserves an entire book dedicated to it.

One (perhaps subtle) difficulty is that it's tough to define what is meant by creativity. For example, in

Treffinger, D. J., Young, G. C., Selby, E. C., & Shepardson, C. (2002). Assessing Creativity: A Guide for Educators. National Research Center on the Gifted and Talented.

the authors write:

Treffinger (1996) reviewed and presented more than 100 different definitions from the literature. Aleinikov, Kackmeister, and Koenig (2000) offered 101 contemporary definitions from children and adults (p. 5).

Furthermore, there is a debate about whether or not creativity is domain-specific (i.e., whether one can be generally creative or whether one should instead use words like mathematically creative).

Rather than attempt to review the creativity literature in its entirety, let me hit a few high points:

$1.$ Creativity research did not take off until about 1950, when the president of the APA, J.P. Guilford, gave his inaugural address (aptly) titled Creativity, in which he called for the concept to be investigated - primarily in terms of various traits.

$2.$ Preceding Guilford's address, related work came from the literature on thought. Wallas' (1926) book The Art of Thought is the first to put forth the five-stage model of problem solving: preparation, incubation, intimation, illumination, and verification.

$3a.$ There are numerous descriptions of mathematical creativity that accord with the incubation model described by Wallas. The most famous example is Poincare stepping onto an omnibus and having a revelation about Fuschian functions; this is contained in his work The Foundations of Science in a chapter (again, aptly) titled Mathematical Creation. This book, and that chapter in particular, encouraged another mathematician to write his own book on mathematical creativity: Jacques Hadamard's The psychology of invention in the mathematical field. (This is all suggested reading!)

$3b.$ For other examples, see Cohen's write-up of his development of forcing (related to my MO answer here) or Gauss' comment about post-incubation illumination, cited in Klamkin's (1994) Mathematical Creativity in Problem Solving and Problem Proposing II as recorded by K. Knopp:

...all the brooding, all the searching was for nothing; finally, a few days ago I succeeded. But not by long searches but by the sheer grace of God, I may say, like lightning strikes, the riddle was solved; I myself would not be able to find the connection between what I knew previously, with what I used for my last attempt and with what finally succeeded.

The lesson of $3a$ and $3b$ is that even great mathematicians are (often) metacognitively unaware as to what produced their final creative insight. What's clear, though, is that before the incubation period, one needs to work consciously on a problem (Wallas' first stage). To quote Poincare:

There is another remark to be made about the conditions of... unconscious work: it is possible, and of a certainty it is only fruitful, if it is on the one hand preceded and on the other hand followed by a period of conscious work... Perhaps we ought to seek the explanation in the preliminary period of conscious work which always precedes all fruitful unconscious labor.

In a related vein, George Polya writes (semi-humorously):

Past ages regarded a sudden good idea as an inspiration, a gift from the gods. You must deserve such a gift by work, or at least a fervent wish.

(For a very outdated and, in my opinion, poorly written article that also emphasizes work, see Adler's (1984) Mathematics and Creativity in Mathematics, People, Problems, Results v2.)

$4a.$ Your question asks: "How can educators encourage creativity without sacrificing correctness?" I think the key is to conceive of creativity as produced by conscious work. One account in the creativity literature to this end is provided by Howard Gruber; see his book Darwin on Man, or compare the Gauss quotation above (with the frequently used metaphor of lightning) to the excerpt here. Much of Gruber's work helps to demystify creativity; to this end, helping students see that even great theoretical advances are a product of hard work and not some magical power can encourage innovation and creativity.

$4b.$ For a concrete suggestion as to how educators can encourage creativity, I would suggest incorporating problem posing (not just problem solving) into the classroom. This relates to creativity and the literature on problem finding; you can find a more extended discussion on the relation between the two in the section Problem posing as a feature of creative activity or exceptional mathematical ability in:

Silver, E. A. (1994). On Mathematical Problem Posing. For the learning of mathematics, 14(1), 19-28.

Sources relating the above work to creativity can be found here, e.g.,

Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(3), 75-80.

Finally, as far as problem posing, the classic text is Brown and Walter's "The Art of Problem Posing," which provides a very accessible account of how to begin with a given mathematical scenario and use it to pose increasingly complex problems, as well as how this can be incorporated into an actual course.

This is tangential, but I'll submit it as an answer because I cannot comment yet.

the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way

It shouldn't be controversial to want the right answer (to an unambiguous question). The 'right way', however, presents serious problems. The 'right way' mentality teaches students that math class is an exercise in memorizing and executing a flow chart. This discourages students from actively thinking about what they're doing, and in turn greatly reduces their chances at becoming creative mathematical thinkers. In order to encourage creative thinking about mathematics, we really just need to encourage thinking about mathematics rather than doing recall of the flow chart.

I've had success in junior and senior high school aged students by challenging their flow-chart operations on extremely simple questions (what this means will vary from student to student). For example, I've seen many students approach equations such as $\frac{x}{2} = 5$ by drawing the $1$ into $\frac{x}{2} = \frac{5}{1}$ so that they can execute 'cross multiplication'. The same people might successfully find a common denominator and add $\frac{5}{1} + \frac{3}{2}$ but fail to do so with $5 + \frac{4}{3}$. These sorts of problems are excellent litmus tests for people executing flow-charts rather than thinking.

To challenge the flow chart, one fruitful place to start is the concept that doing something to both sides of an equation yields a new equation which is exactly as true as the previous one. When a student is on board with this (as in, actually has an intuitive understanding of its truth), they're not far from putting cross-multiplication behind them.

This must be done with prodding questions rather than instruction, which is time consuming and, to be honest, a little mentally taxing (trying to find just the right question for student X in moment Y is tough). It's worth it.

Ask students to solve problems, pose their own questions, make their own definitions, etc... This is about creating opportunity for creative work.

But it is also crucial that you then take the student's ideas seriously! Give the ideas respect. Encourage other students to do the same. Don't replace student thinking or explanation with your own, instead ask questions to help the student refine their ideas.

Students need a safe environment (intellectual and psychological) so that they can feel free to take risks. Then you can give them opportunity and freedom. But unless they feel respected and comfortable, you won't get anywhere.

• Do you mean that if a student makes a wrong assumption such as 0.999... $\neq$ 1, it's probably better if the teacher instead of telling the student that that's incorrect either shows the student a derivation of a contradiction if they can based an another of the student's assumptions or teaches the formal meaning of the statement 0.999... = 1 so that they cannot question it? If the student says the solution to the problem is that a real number is a decimal representation but has an open mind about addition of 0.333... not being bijective or things that infinite integers exist like I did but – Timothy Dec 27 '18 at 18:22
• also things that only those notations that eventually terminate whether at finite or infinite position actually represent a number and those with a nonzero digit at the position for every number no matter how large don't represent a number, then they might think that infinitesimal numbers exist but multiplication by 3 is not bijective so the number $\frac{1}{3}$ doesn't so the solution might be to teach them that the real numbers are those constructed a certain way. – Timothy Dec 27 '18 at 18:27
• Depends upon the situation. But my first instinct is to start asking such a student questions. I first want to know what they think is true and why. At some point in the conversation, which could take a long time, I would hope to transition to asking questions that the student could work out and use to see why we often say those two things are "the same" real number. – TJ Hitchman Jan 3 '19 at 19:48

My brief answer would include: interdisciplinary, integration with arts, open-ended problems. Of course, right answers are important as long as we can nurture the idea that not every question has one perfect answer. Instead of What is 2+3? I like asking children to find all dominoes that have total of 5 dots. I ask the same questions future teachers, broadening the domain ...

It helps getting students interested in some of the more "recreational mathematics" type sites and problems. Solving some not-completely-serious problem that interests them is as much creative and intellectually demanding (and often less demanding in specific background, a impassable roadblock for beginners) than "serious" work.

Something that helps, is to create problems where it's harder to see the "right way" to solve it. I try to give problems to my students where it's not obvious what type of method may be best to solve them, and then we discuss whatever solutions that come up.

I also ask them to solve problems in more than one way. The idea, being that if they were teaching someone else, and their student didn't understand the first method, they would have to find another.

Edit: Included the following after extremely helpful suggestions from others

Sorry! Here is one: http://illuminations.nctm.org/Lesson.aspx?id=1037 Here is a lesson that I often do with my students from Grade 6 - 8. I try to have students work in groups of ~4 and try to come up with at least 2 ways to solve this. It's also interesting when they see the problem again the following year, and have to try to solve it using something new that we have learned that year. Then, we start with what we remember from the previous year, before trying to find a new method.

• This is essentially what Pólya's "How to solve it" suggests – vonbrand Aug 3 '15 at 14:57

I agree that "the emphasis on mathematics education is typically on getting the right answer, and often in doing it the right way". I also agree on many of the points made in the previous answers but I am a bit surprised that one point was entirely missed: Given that, these days, mathematics exposition is by short topics without much of a "story line", it is just about impossible for a student to be in a position to be creative. To be creative, it is necessary to have experienced how an idea that worked in a particular context can also work in an apparently completely different context.

Here is what Thurston had to say about something very close to it:

Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics. (Thurston, W. P. (1990). Mathematical Education, Notices of the American Mathematical Society, 37 7, 844-850)

But, at least for the beginners, that joy requires that the contents in the materials available to them be themselves tightly conceptually organized to show the "story line"---as opposed to being a collection of topics. But such materials, sadly, are very much lacking.

Paul Lockhart addresses these issues specifically at the high school and early university level mathematics in Measurement. His work is significant in that his writing itself encourages creativity, rather than giving advice on how to encourage creativity.

Lockhart does this to a lesser extent in A Mathematician's Lament (here: a free pdf earlier version of this text ).

I gave a talk a couple years ago that tried to synthesize my experience and reading. http://bit.ly/goldenggb13 (Also on GeoGebra)

The two main pools of ideas for me in practice come via Ken Robinson and the MIT Media Lab, especially Mitch Resnick (whose work follows the great Seymour Papert).

Sir Ken: creativity is ideas with value. “Teaching for creativity involves asking open-ended questions where there may be multiple solutions; working in groups on collaborative projects, using imagination to explore possibilities; making connections between different ways of seeing; and exploring the ambiguities and tensions that may lie between them.” - Sir Ken Robinson, Out of Our Minds

Resnick: creativity in practice is a spiral: imagine -> create -> play -> share -> reflect --> imagine...

Students need an opportunity for choice and an opportunity to make. Teacher provides a safe place, constructive feedback, and -hopefully - connects the students with an authentic audience.