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Connecting critical thinking skill & Fermi problems

By Frank Witte, on 19 November 2016

Critical thinking usually requires the application of skill and knowledge from a wide range of different fields. It also demands making use of reasonable guesses and estimates of things you don’t know. Our media sometimes play risky games, for entertainment, with their credibility. A Supermoon example and the case for teaching Fermi problems.

The Supermoon of 14/11/2016

Hardly anyone who has opened a newspaper, watched some TV or checked out their social-media timelines in the week of November the 14th of 2016 can have missed the reporting on the ‘Supermoon’.SuperMoon A Supermoon [1] is a rather arbitrarily defined ‘thing’ that, on the basis of the Science behind it, is neither of any scientific interest nor in any other way significantly affects us or our world. That changes however due to the way it s being reported as a ‘Science interest’ story. But in an age of widespread scepticism among the electorate in western countries about anthropogenic global climate change we certainly are very careful about how we report Science, right? Wrong!

The distance of the Moon to the Earth is not constant and it varies along the Moon’s orbit by about 12% relative to the largest distance (apogeum). ‘Supermoons’ roughly arise when, within an hour around the ‘Full Moon’ phase, the Moon is closer then 90% of the apogeum distance. Of course the Moon is closer to the Earth for some time every 27 days, but then its phase of illumination isn’t ‘full’. In terms of Science it is a wholly arbitrary definition in order to make something into an ‘event’.

What makes this event interesting is the way it is being reported and how people respond to it. In the days after the last Supermoon postings of the above picture on social media by plenty of people drew a slew of comments from other people. The tenor of those comments was “Don’t spoil it!”. However spoiling it is an excellent example of how some training with ‘Fermi problems’ can actually empower people to do some critical thinking.

Fermi problems & Critical Thinking

Fermi problems are named after the twentieth century, Italian, all-round physicist Enrico Fermi. It is a category of problems where relatively simple quantitative analysis allows one to come up with reasonable guesses or estimates for the solution [2]. A typical example I like using in my classes is to challenge my economics students to estimate how many piano-tuners there are in London without consulting the yellow pages or branch organisations. What they learn in doing such problems is how to creatively connect a set of different things they know about London, with estimates that seem reasonable based upon their own experience with London and with one another using simple arithmetic.

In the ‘London Piano-tuners’ case they could use estimates of population-size, typical annual living wage in London, typical household size, probability of a household having a piano, typical frequency with which a piano needs tuning, typical cost of a tuning session, etc, to arrive at an estimate of the number of full-time piano-tuner equivalent. The nicest aspect is to confront that result then with a statistic coming from the Yellow Pages or a branch organisation. The purpose of Fermi problems is to illustrate that by making reasonable guesses and applying common sense we can get the actual fact right within a reasonable margin (reasonable relative to the raw estimates and simplifications used).

Doing such problems trains students in comparing their common sense with real-world data, in making assessments such as ‘how wrong’ an estimated number is likely to be, in recognizing that errors in multiple guesses have a way of averaging out when they are not biased. Finally, and maybe most importantly, it empowers them with a skill that allows them to do common sense and reasonability checks on ‘facts’ or ‘opinions’ [3] hurled at them in the media, social media and of course also in lecture theatres.

Geometry of a Supermoon

The last Supermoon was widely advertised in the media and online as a phenomenon that was not to be missed. Who knows whether you would be around for the next Supermoon? This was typically illustrated by the number that a Supermoon would appear 15% bigger and 30% brighter! And sure, plenty of people who normally don’t spend much time observing the Moon went out in the early evening (if no clouds)  to see the Moon rise and probably thought ‘Wow, that one is big!’ However equally many people went out a few hours later and felt compelled to agree even though there was this nagging feeling something was wrong. So what is happening here?

First of all the 15% number is rather misleading because it is the maximal change in apparent size when you would have a full Moon in closest to Earth compared to a full Moon farthest from Earth. When we compare the Earth-Moon distance of the last Supermoon with the average Earth-Moon distance we are talking about a 6.3% difference.  This roughly translates into a 6.3% difference in angular diameter of the Moon as seen from the Earth. This is about the difference the picture in this post shows!

This 6.3% difference should however be compared to other ‘distance related’ apparent size-differences affecting the Moon’s appearance. For example an observer who sees the Moon on the horizon is geometrically farther away from the moon then an observer who, at the same time, sees the Moon overhead because the Earth is a ball. The magnitude of this effect is about 1.5% of the Moon’s apparent diameter. So purely geometrically, people who saw the supermoon on the horizon last week saw only a 4.8% increase … but only if they were using a measurement device!

Perception of a Supermoon

However the vast majority of people weren’t using measurement devices but where relying on their ‘visual perception’ of the size. Yet the human visual perception of a Moon close to the horizon is subject to the so-called Moon illusion [4]! Empirical data shows that humans perceive a Moon close to the horizon on average as about 50% larger than that same Moon overhead. Note that this happens despite the fact the Moon’s apparent diameter is geometrically actually 1.5% smaller at that moment. In other words, perception anomalies are more than a factor 10 larger as an effect shaping our impression of the Supermoon than the geometric effects related to the Moon’s orbit that define the Supermoon.

This makes it reasonable to assert that a ‘Supermoon’ like last week is, astronomically and in terms of the visual experience, a non-event. All that makes it an event is the media and social-media hype surrounding it.  Hype=Event.


So what about all those pictures illustrating the ‘Supermoon’? This is where the story turns a little darker. These picture depict something the photographers (or photo-editors) do not actually visually experience. Most are made by taking tele-lens shots of distant landmarks in the same frame as the full Moon. This is called a ‘forced perspective’, a well-know film technique used to create an artificial size-difference between objects in the same frame. Unfortunately not only was the actual geometric supermoon effect invisible to casual , naked-eye, Moon observers last week. But those producing the illustrative picture to go along with this hyped non-event had to deliberately choose a point-of-view to confirm people that they had seen something they had not seen.

To few people in our society are trained to approach news reporting and ‘facts’ stated in debates as Fermi problems that are an open invitation for common sense and reasonability checks. But most people are not fools and when they’re told they are seeing something that they’re not seeing … there is for many a gut feeling that tells them so. The otherwise harmless Supermoon hypes contribute to this. But people who are untrained in critically examining facts and numbers, as if they were Fermi problems, have no other resort than to be sceptical on the basis of the perception of being taken for a fool. People who are trained to examine such hyped claims, which hopefully includes all our students, face a choice between ‘spoiling it’ or revealing it for what it is.

Societal impacts

I have not tried, as a Fermi problem, to see what common sense or reasonable estimates would say about the impact of such hyped non-events on public perception of Science outcomes. But in a world in which it is crucial that people retain some confidence in Science reporting about issues such as anthropogenic climate change, media and social media should perhaps be less eager to fool and mislead on something as trivial as an otherwise unremarkable full Moon? Equipping people with the arithmetic skills to solve such Fermi problems goes some way towards empowering them to ‘see through’ such hypes themselves. However the arithmetic skills themselves are not enough! The skill of juggling numbers you know from a reliable source, the reliability of sources, the skill to make reasonable estimates of things you don’t know and finally the skill of being able to critically look at your outcomes and their validity are equally essential. Fermi problems encompass all those dimensions when taught properly. Not teaching this properly might just really be about “spoiling it”!


[1] https://en.wikipedia.org/wiki/Supermoon

[2] Ärlebäck, J.B., 2009. On the use of realistic Fermi problems for introducing mathematical modelling in school. The Mathematics Enthusiast, 6(3), pp.331-364; Ärlebäck, J.B. and Bergsten, C., 2013. On the use of realistic Fermi problems in introducing mathematical modelling in upper secondary mathematics. In Modeling Students’ Mathematical Modeling Competencies (pp. 597-609). Springer Netherlands; Sriraman, B. and Lesh, R.A., 2006. Modeling conceptions revisited. ZDM, 3 (3), pp.247-254; Peter-Koop, A., 2009. Teaching and Understanding Mathematical Modelling through Fermi-Problems. In Tasks in primary mathematics teacher education (pp. 131-146). Springer US.

[3]  Lunt, B.M. and Helps, C.R.G., 2001. Problem solving in engineering technology: Creativity, estimation and critical thinking are essential skills. In ASEE Annual Conference & Exposition Paper ID (Vol. 803, pp. 1-9); Sriraman, B. and Knott, L., 2009; The mathematics of estimation: Possibilities for interdisciplinary pedagogy and social consciousness. Interchange, 40(2), pp.205-223; Sharma, M.B. and Elbow, G.S., 2000. Using internet primary sources to teach critical thinking skills in geography. Greenwood Publishing Group; Pierce, C.E., Gassman, S.L. and Huffman, J.T., 2013. Environments for fostering effective critical thinking in geotechnical engineering education (Geo-EFFECTs). European Journal of Engineering Education, 38(3), pp.281-299; Efthimiou, C., Llewellyn, R., Maronde, D. and Winningham, T., 2006. Physics in Films: an assessment. arXiv preprint physics/0609154.

[4] Kaufman, L. and Kaufman, J.H., 2000. Explaining the moon illusion. Proceedings of the National Academy of Sciences, 97(1), pp.500-505; von Békésy, G., 1949. The moon illusion and similar auditory phenomena. The American journal of psychology, 62(4), pp.540-552.


3 Responses to “Connecting critical thinking skill & Fermi problems”

  • 1
    Alarcos wrote on 19 November 2016:

    RT @UCLConnectedC: Connecting critical thinking skill & Fermi problems: Critical thinking usually requires the appl… https://t.co/TeTXora…

  • 2
    Brent Carnell wrote on 22 November 2016:

    Critical thinking, questioning facts, learning through enquiry, debating with fellow students and with staff – these are all touched on in this post, which are also core to the UCL Connected Curriculum. Social media is changing the production of knowledge (and flawed knowledge – just last week 1000s of people believed the inaccurate news that Mark Zuckerberg was turning to a political career[1]!); let’s hope it’s also bringing criticality into our everyday lives! Whether that is rigorous, researched, or constructive, or even without name-calling, is another story!

    [1] http://www.bbc.co.uk/news/technology-38036730

  • 3
    Liana wrote on 5 August 2022:

    thanks for info

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