Tag Archives: problem-solving

Puzzles and mysteries

Detail from abstract by Zahrah ReshPuzzles and mysteries are a pair of words that have taken on a whole new meaning for me since reading John Kay’s and Mervyn King’s book called ‘Radical uncertainty: decision-making for an unknowable future‘ during the summer vacation [see ‘Where is AI on the hype curve?‘ on August 12th, 2020]. They describe puzzles as well-defined problems with knowable solutions; whereas mysteries are ill-defined problems, that have no objectively correct solution and are imbued with vagueness and indeterminacy.  I have written before about engineers being creative problems-solvers [see ‘Learning problem-solving skills‘ on October 24th, 2018] which leads to the question of whether we specialise in solving puzzles or mysteries, or perhaps both types of problems.  The problems that I set for students to solve for homework to refine and evaluate their knowledge of thermodynamics [see ‘Problem-solving in thermodynamics‘ on May 6th, 2015] clearly fall into the puzzle category because they are well-defined and there is a worked solution available.  Although for many students these problems might appear to be mysteries, the intention is that with greater knowledge and understanding the mysteries will be transformed into mere puzzles.  It is also true that many real-world mysteries can be transformed into puzzles by research that advances the collective knowledge and understanding of society.  Part of the purpose of an engineering education is to equip students with the skills to make this transformation from mysteries to puzzles.  At an undergraduate level we use problems that are mysteries only to the students so that success is achievable; however, at the post-graduate level we use problems that are perceived as mysteries to both the student and the professor with the intention that the professor can guide the student towards a solution.  Of course, some mysteries are intractable often because we do not know enough to define the problem sufficiently that we can even start to think about possible solutions.  These are tricky to tackle because it is unreasonable to expect a research student to solve them in limited timeframe and it is risky to offer to solve them in exchange for a research grant because you are likely to damage your reputation and prospects of future funding when you fail.  On the other hand, they are what makes research interesting and exciting.

Image: Extract from abstract by Zahrah Resh.

Meta-knowledge: knowledge about knowledge

As engineers, we like to draw simple diagrams of the systems that we are attempting to analyse because most of us are pictorial problem-solvers and recording the key elements of a problem in a sketch helps us to identify the important issues and select an appropriate solution procedure [see ‘Meta-representational competence’ on May 13th, 2015].  Of course, these simple representations can be misleading if we omit parameters or features that dominate the behaviour of the system; so, there is considerable skill in idealising a system so that the analysis is tractable, i.e. can be solved.  Students find it especially difficult to acquire these skills [see ‘Learning problem-solving skills‘ on October 24th, 2018] and many appear to avoid drawing a meaningful sketch even when examinations marks are allocated to it [see ‘Depressed by exams‘ on January 31st, 2018].  Of course, in thermodynamics it is complicated by the entropy of the system being reduced when we omit parameters in order to idealise the system; because with fewer parameters to describe the system there are fewer microstates in which the system can exist and, hence according to Boltzmann, the entropy will be lower [see ‘Entropy on the brain‘ on November 29th, 2017].  Perhaps this is the inverse of realising that we understand less as we know more.  In other words, as our knowledge grows it reveals to us that there is more to know and understand than we can ever hope to comprehend [see ‘Expanding universe‘ on February 7th, 2018]. Is that the second law of thermodynamics at work again, creating more disorder to counter the small amount of order achieved in your brain?

Image: Sketch made during an example class

Knowledge explosions

Photo credit: Tom

When the next cohort of undergraduate students were born, Wikipedia had only just been founded [January 2001] and Google had been in existence for just over a decade [since 1998].  In their lifetime, the number of articles on Wikipedia has grown to nearly 6 million in the English language, which is equivalent to 2,500 print volumes of the Encyclopedia Britannica, and counting all language editions there are 48 million articles.  When Leonardo Da Vinci was born in 1452, Johan Gutenberg had just published his first Bible using moveable type.  By the time Leonardo Da Vinci was 20 years old, about 15 million books had been printed which was more than all of the scribes in Europe had produced in the previous 1500 years.  Are these comparable explosions in the availability of knowledge?  The proportion of the global population that is literate has changed dramatically from about 2%, when Leonardo was alive, to over 80% today which probably makes the arrival of the internet, Wikipedia and other online knowledge bases much more significant than the invention of the printing press.

Today what matters is not what you know but what you can do with the knowledge because access to the internet via your smart phone has made memorisation redundant.

Learning problem-solving skills

Inukshuk: meaning ‘in the likeness of a human’ in the Inuit language. A traditional symbol meaning ‘someone was here’ or ‘you are on the right path’.

One definition of engineering given in the Oxford English Dictionary is ‘the action of working artfully to bring something about’.  This action usually requires creative problem-solving which is a common skill possessed by all engineers regardless of their field of specialisation.  In many universities, students acquire this skill though solving example problems set by their instructors and supported by example classes and, or tutorials.

In my lectures, I solve example problems in class using a pen and paper combined with a visualiser and then give the students a set of problems to solve themselves.  The answers but not the solutions are provided; so that students know when they have arrived at the correct answer but not how to get there.  Students find this difficult and complain because I am putting the emphasis on their learning of problem-solving skills which requires considerable effort by them.  There are no short-cuts – it’s a process of deep-learning [see ‘Deep long-term learning’ on April 18th, 2018].

Research shows that students tend to jump into algebraic manipulation of equations whereas experts experiment to find the best approach to solving a problem.  The transition from student to skilled problem-solver requires students to become comfortable with the slow and uncertain process of creating representations of the problem and exploring the possible approaches to the solution [Martin & Schwartz, 2014].  And, it takes extensive practice to develop these problem-solving skills [Martin & Schwartz, 2009].  For instance, it is challenging to persuade students to sketch a representation of the problem that they are trying to solve [see ‘Meta-representational competence’ on May 13th, 2015].  Working in small groups with a tutor or a peer-mentor is an effective way of supporting students in acquiring these skills.  However, it is important to ensure that the students are engaged in the problem-solving so that the tutor acts as consultant or a guide who is not directly involved in solving the problem but can give students confidence that they are on the right path.

[Footnote: a visualiser is the modern equivalent of an OverHead Projector (OHP) which instead of projecting optically uses a digital camera and projector.  It’s probably deserves to be on the Mindset List since it is one of those differences between a professor’s experience as a student and our students’ experience [see ‘Engineering idiom’ on September 12th, 2018]].


Martin L & Schwartz DL, A pragmatic perspective on visual representation and creative thinking, Visual Studies, 29(1):80-93, 2014.

Martin L & Schwartz DL, Prospective adaptation in the use of external representations, Cognition and Instruction, 27(4):370-400, 2009.


CALE #9 [Creating A Learning Environment: a series of posts based on a workshop given periodically by Pat Campbell and Eann Patterson in the USA supported by NSF and the UK supported by HEA] – although this post is based on an introduction to tutorials given to new students and staff at the University of Liverpool in 2015 & 2016.

Photo: ILANAAQ_Whistler by NordicLondon (CC BY-NC 2.0) https://www.flickr.com/photos/25408600@N00/189300958/