Problems are bad. This is a common-sense consensus that can
be easily drawn from our life’s common experiences. However, a person that
possesses the ability to solve problems can easily thrive in life. The
discussion here will show how significant Problem-Solving skills are, both in
academic and professional environments; this, in turn, will show that it is
possible to use problems to our benefit.

We can define Problem Solving as the threefold ability to analyze
complex problems and generate creative solutions supported by informed evidence.
It can be derived from this definition that the complimentary skills of Critical
Thinking, Creative Thinking and Information Literacy are to be adequately
understood so that one can apply Problem Solving in an adequate manner.

A question that can come from this definition is why one
should adopt the characteristics of Problem Solving as outlined by the definition
above? To answer that, we can traverse through some other definitions of
Problem Solving. Two of the three aspects – Information Literacy and Critical
Thinking are found in some definitions (Cummins, Yamashita, Millar, & Sahoo, 2019) ; (Symons & Pierce, 2019) , (Mohaghegh & Größler, 2020) , but none of them
brings the aspect of Creative Thinking to the discussion.

If the benefit of adding this third aspect to the table can
be shown, it can be inferred that this broader definition can have an added
benefit, if applied. Creative Thinking can be understood as the ability to
synthesize and connect multiple ideas, whether convergent or alternate, in an
innovative way, with a willingness to take risks. It is left to determine
whether this understanding that Creative Thinking should be considered
advantageous to students.

In order to introduce this discussion, it is relevant to
outline a view of Creativity/Creative Thinking that allows to justify its
inclusion within the definition of Problem-Solving cited above. A short review
of some studies reveal that they tend not to define creativity (Keenan-Lechel & Henriksen, 2019) , (Henriksen & Mishra, 2020) , (Cain & Henriksen, 2017) ; however, two
definitions fit the analytical context of this essay: one portrays creativity
as “an approach to problem-solving through the imagination” (Evans, Henriksen, & Mishra, 2019) and another that
views it as “the genesis of all learning, in every area and across every
discipline” (Richardson, Henriksen, & Mishra, 2017) .

*per se*, but instead they discuss aspects related to creativity
This shows that in a scholarly point of view, at least, it
is possible to see creativity as a fundamental skill linked to problem-solving.
However, to complete the effort, it is necessary to verify if this view is
shared in the professional environment as well.

There was an extensive discussion at the World Economic
Forum in 2015 about the skills that are needed by workers in the following 5
years. A list was then compiled, containing the top 10 skills workers should
possess in order to thrive in the market. The first three skills listed are
Complex Problem Solving, Critical Thinking and Creativity (World Economic Forum, 2016) . Creativity was
added because of the understanding that it would be necessary in order to
adequately deal with the innovative new technologies.

So, from this it is possible to see that in both
environments – academic and professional – the role of Creativity/Creative
Thinking in Problem-Solving is very relevant. Two practical examples of how
problems end up being used for added benefit, with a creative approach, are
shown in two famous situations: Fermat’s Last Theorem and the Clausewitzean
Theory of War.

It is said that about 1637, a French lawyer called Pierre de
Fermat wrote 35 words in the margin of his copy of Diophantus’s

*Arithmetica*, unknowingly causing a ruckus that would last for almost four centuries:*Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.[1]*

*(Dixon, 1920)*

There are still controversies about whether he did solve
this or not, since no actual proof was found among his writings save it for (Dixon, 1920) .
One of these attempts is an example of how problems, even when unsolved, can be
useful: Ernst Eduard Kummer, around the 1840s, after several unsuccessful
attempts to demonstrate the theorem, ended up creating a definition for ideal
numbers.

*n = 4*, but for the general problem that became known as*Fermat’s Last Theorem*, the challenge had just begun. Hundreds of attempts to demonstrate the theorem were made in the subsequent 350 years
From that definition arose the general theory of algebraic numbers,
one of the most important branches of modern mathematics (Bell, 1986) .

The beautiful demonstration of this theorem began when in
1984, Gerhard Frey hypothesized about a link between two apparently unrelated
problems: the Taniyama-Shimura conjecture and the Fermat’s Last Theorem (Singh, 2013) . This inspired an
English mathematician, that managed to formally prove the theorem in 1995, 358
years after its conjecture (Knudsen, 2015) , (Wiles, 1995) .

#
References

Bell, E. T. (1986).

*Men of Mathematics.*New York: Simon and Schuster.
Cain, W., & Henriksen, D. (2017). Uncreativity:
a Discussion on Working Creativity Before and After Ideation with Dr. Chris
Bilton.

*TechTrends, 61*, 101-105.
Cummins, P. A., Yamashita, T., Millar, R. J., &
Sahoo, S. (2019, August). Problem-Solving Skills of the U.S. Workforce and
Preparedness for Job Automation.

*Adult Learning*, pp. 111-120.
Dixon, L. E. (1920).

*History of the Theory of Numbers Vol. 2.*Washington: Carnegie Institution.
Evans, M. D., Henriksen, D., & Mishra, P.
(2019). Using Creativity and Imagination to Understand our Algorithmic World:
a Conversation with Dr. Ed Finn.

*TechTrends, 63*, 362-368. doi:10.1007/s11528-019-00404-3
Henriksen, D., & Mishra, P. (2020). A Pragmatic
but Hopeful Conception of Creativity: a Conversation with Dr. Barbara Kerr.

*TechTrends, 64*, 195-201. doi:10.1007/s11528-020-00476-6
Keenan-Lechel, S. F., & Henriksen, D. (2019).
Creativity as Perspective Taking: An Interview with Dr. Vlad Glăveanu.

*TechTrends, 63*, 652-658. doi:10.1007/s11528-019-00437-8
Knudsen, K. (2015, August 20).

*The Math Of Star Trek: How Trying To Solve Fermat's Last Theorem Revolutionized Mathematics*. Retrieved from Forbes: https://www.forbes.com/sites/kevinknudson/2015/08/20/the-math-of-star-trek-how-trying-to-solve-fermats-last-theorem-revolutionized-mathematics/amp/
Mohaghegh, M., & Größler, A. (2020). The
Dynamics of Operational Problem-Solving: A Dual-Process Approach.

*Systemic Practice and Action Research*, pp. 27-54.
Richardson, C., Henriksen, D., & Mishra, P.
(2017, September 6). The Courage to be Creative: An Interview with Dr. Yong
Zhao.

*TechTrends, 61*, pp. 515-519. doi:10.1007/s11528-017-0221-1
Singh, S. (2013).

*The Simpsons and Their Mathematical Secrets.*A&C Black.
Symons, D., & Pierce, R. (2019). Beyond fluency:
Promoting mathematical proficiencies through online collaborative problem
solving.

*Australian Primary Mathematics Classroom*, pp. 4-8.
Wiles, A. (1995). Modular elliptic curves and
Fermat's Last Theorem.

*Annals of Mathematics, 142*, 443-551. doi:10.2307/2118559
World Economic Forum. (2016, January 19).

*The 10 skills you need to thrive in the Fourth Industrial Revolution*. Retrieved from World Economic Forum: https://www.weforum.org/agenda/2016/01/the-10-skills-you-need-to-thrive-in-the-fourth-industrial-revolution/
[1]
[It is not possible to divide]

*a cube into two cubes, or a fourth power into two fourth powers, or in general, until infinity, above fourth power into two similar powers. I have found a wonderful demonstration for this, but it won’t fit in this margin.*In more modern words, it is to say that there are no three positive integers*a, b*and*c*that satisfy the equation*a**+*^{n}*b**=*^{n}*c**for any integer value of*^{n}*n*greater than 2 (free translation by the author, with terms added for clarification).
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