Jonathan Beale, Researcher-in-Residence, CIRL
Yesterday, CIRL inaugurated its colloquia series, where we invite leading experts in education to speak on important themes in teaching and learning. Our first speaker was educator and author Ian Warwick, whose most recent book, Learning with Leonardo: Unfinished Perfection (John Catt, 2019), co-written with Ray Speakman, offers a study of the ways in which Leonardo da Vinci inquired and learned, and outlines several lessons we can learn from Leonardo about teaching and learning.
Leonardo is the archetype of an intellectually curious human being. He was relentlessly and tirelessly fascinated by all things. This motivated him to make detailed inquiries into the most specific details of his work. For example, yesterday, Warwick described how, when painting the Mona Lisa, Leonardo would sneak into morgues at night to dissect the faces of corpses in order to gain a greater understanding of what happens within the human face when we smile.
Leonardo, Warwick and Speakman write, possessed the curiosity of a child, perpetually looking about his surroundings and asking, ‘Why?’. They describe Leonardo as ‘the most relentlessly questioning man in history’ (Learning with Leonardo, p. 33). His state of mind was more than just intellectual curiosity; it was a childlike sense of wonder which he maintained his entire life.
Warwick and Speakman argue that teachers should aim to inspire such wonder in students. We should also seek to elicit such wonder in our peers and ourselves. ‘We should be careful’, they argue, ‘to never outgrow our wonder years’ (ibid.). Wonder can motivate us to pursue answers to questions and solutions to problems; make the kinds of inquiries that yield new discoveries; identify new connections; draw novel interpretations; and see things in hitherto unseen ways.
How do we elicit wonder? An important first stage is to inspire intellectual curiosity, which can develop into wonder. This curiosity need not be specific and targeted; it can be a general state of mind. Leonardo was curious about everything. His curiosity was not always ordered and focused; it would begin as a ‘rampant diversive curiosity’ which would become ‘reined in by dogged investigation and experimentation, to a point where it becomes more specific and focused’ (Learning with Leonardo, p. 34). Leonardo’s diversive curiosity would often become channelled and focused, allowing him to home in on specific details, such as the attention to detail in the Mona Lisa’s smile.
‘Diversive curiosity’ was arguably vital to the polymathic character Leonardo cultivated. Only through possessing such wide-ranging interests could Leonardo have developed knowledge of such a remarkably wide array of fields, some of which appear on the surface to be disparate, ranging from art to astronomy. His thoughts would cover so many subjects in his notebooks that he offered apologies to the reader: ‘Do not blame me, reader, because the subjects are many’ (Codex Arundel; quoted in Learning with Leonardo, p. 35). Yet this wide intellectual peregrination was vital to the formation of his genius: that of a polymath able to see connections throughout all things invisible to almost all other people, such as the mathematical connections between the human body and the universe illustrated in the Vitruvian Man (c. 1490).
If such curiosity is developed and harnessed, it can be pedagogically useful and intellectually virtuous. If not, it can hamper learning. ‘General inquisitiveness’, Warwick and Speakman write in another of their books, Defining More Able Education (Routledge, 2018), ‘sometimes ignites learning, sometimes distracts from learning’. ‘The curious student can be wonderful to teach, the curious student whose interests are unchanneled can become a distraction’ (Defining More Able Education, p. 131). If not harnessed, general curiosity can yield a shallow and fleeting fulfilment of a craving – the kind we experience when we scroll through social media feeds and click on a link that looks interesting, if only to read a couple of sentences and move on.
Diversive curiosity can develop into something more powerful: wonder. Warwick and Speakman write that ‘Curiosity engages, leading to the kind of wonder that both inspires and challenges’ (Learning with Leonardo, p. 39). What kind of wonder should we engender in students, and how can it best be utilised?
A way of inspiring wonder can be found in that discipline that has been said, since its inception on the streets of Athens, to begin with wonder: philosophy. As Socrates remarks in Plato’s Theaetetus:
[W]onder is the feeling of a philosopher, and philosophy begins in wonder [thaumazein]. – Plato, Theaetetus 155c-d (c. 369 BCE)
Plato uses the Ancient Greek word ‘thaumazein’ (θαυμάζειν), which has the connotation of being astonished or perplexed. Philosophy, according to its Socratic origins, begins with such a feeling.
The philosophical method inaugurated by Socrates aims to find the definition of a philosophical concept, such as knowledge, virtue, justice or piety. It begins with Socrates asking his interlocutors probing questions about something they typically take for granted, to engender puzzlement about the concept under scrutiny and give his interlocutors reason to question that which they take for granted.
For example, in Plato’s Sophist, one of Socrates’ interlocutors expresses the following puzzlement about the concept of ‘being’:
For manifestly you have long been aware of what you mean when you use the expression “being”. We, however, who used to think we understood it, have now become perplexed. – Plato, The Sophist, 244a (c. 360 BCE)
Such wonder can inspire and motivate philosophical inquiry. If one does not think there is a question that needs to be raised or a problem that needs to be solved, one will not pursue an answer or a solution. A person is far more likely to engage deeply with a question or problem if it grips them with wonder.
It is to this end that the German philosopher Martin Heidegger used the above passage from the Sophist to motivate his readers to seek an answer to the ‘question of the meaning of being’. The above passage is quoted at the start of the exergue which opens Being and Time (Sein und Zeit, 1927), followed by Heidegger’s opening words,
Do we in our time have an answer to the question of what we really mean by the word ‘being’? Not at all. So it is fitting that we should raise anew the question of the meaning of being.
Few of us stop to ask what we mean by ‘being’. To see that there is a question that needs to be raised, we need to be curious about the concept. One way to do this is to elicit wonder about what it means for something, or anything, to be.
Such wonder is the feeling captured by ‘thaumazein’. As Plato puts it, philosophy not only begins with wonder, but it is the feeling of a philosopher: that which needs to be maintained in order for philosophical inquiries to be undertaken. We need to not grow out of what Warwick and Speakman call our ‘childlike sense of wonder’.
There are many ways the power of thaumazein can be harnessed in the teaching of philosophy. One could begin a lesson with a paradox, such as the liar paradox (‘I am lying’ – if this is true, how can it be?), or one of Zeno’s paradoxes:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. - Zeno, as recounted by Aristotle, Physics, VI:9, 239b15 (c. 350 BCE).
The feeling of thaumazein and the pedagogical opportunities it affords are not confined to philosophy. Wonder, as Leonardo illustrated throughout his life, can motivate one to inquire into all fields.
While Zeno’s paradoxes are philosophical, they are also mathematical conundrums. Another one could be the following, employed by Ludwig Wittgenstein in one of his lectures. Suppose a rope is stretched tightly around the earth at the equator. Suppose the rope is then extended by a yard. If the rope were kept taut and circular in form, how far above the surface of the earth would it be? The natural response is that the distance from the surface of the earth would be infinitesimal. The distance would, however, be nearly six inches.
This surprises us because, given the length of the rope, our intuition is that an extension of one yard would be so insignificant in proportion to the rope’s overall length that it would make barely any difference to its distance from the ground. But the extension of one yard does make barely any difference to the rope’s distance from the ground. Six inches is not the tiniest of distances from the surface of the earth, certainly not the barely perceptible distance we expect to result from the rope’s extension. Nor is it an insignificant proportion of one yard. But these are the wrong comparisons to draw to get the appropriate sense of proportion. Six inches is the tiniest of fractions when considered in relation to the radius of the earth: a fraction of around 1/40million, the same fraction a yard is of the length of the rope. So it is correct that lengthening the rope by a yard would increase the rope’s distance from the earth by a tiny fraction.
In this puzzle, we are misled by comparing the length of the additional piece of rope with the length of the whole. To clear up this conundrum, we just need to get a sense of proportion.
Paradoxes such as Zeno’s and a conundrum such as that raised by Wittgenstein could as much be used to raise wonder in mathematics as in philosophy. Within Wittgenstein’s work on the philosophy of psychology we also find an illusion that could be used to generate wonder in psychology, that of the ‘duck-rabbit’:
Wittgenstein used this to illustrate ‘aspect perception’ and the shift we undergo when we move from seeing one aspect of something to another. While you’re aware that the image can represent both a duck and a rabbit, your visualisation and conceptualisation of the image undergoes a shift when you move from seeing it as a duck to seeing it as a rabbit (or vice versa).
Thomas Kuhn would use this illusion to depict what takes place during a ‘paradigm shift’, where a scientific revolution results in the replacement of one scientific paradigm with another, bringing about a change in our world-view, such that we see the same information but in a completely new way. In that way, the illusion could also be used to inspire wonder in the sciences, by illustrating the ways in which scientific revolutions such as those inaugurated by the discoveries of Copernicus and Darwin have changed our entire way of seeing the world and our place in it.
In psychology, many recent cases of perceptual phenomena could be used to generate wonder. Take, for example, an image that went viral in 2015 and has come to be referred to as ‘the dress’, which some people see as blue and black, others as white and gold:
The fact that some people see the dress as containing different colours can generate wonder. There are other recent, similar, examples such as the shoe that some people see as white and pink, others as green and grey, or the sound clip that some people hear as ‘Yanny’, others as ‘Laurel’.
Intellectual curiosity is important for effective teaching and learning. But if we can inspire something greater, wonder, then the pedagogical benefits can also be greater. One thing we can learn from geniuses like Leonardo is the usefulness of a particular experience for teaching and learning: childlike wonder.
 See Norman Malcolm, Ludwig Wittgenstein, A Memoir (Oxford: Oxford University Press, 2001 ), p. 46.
 See P.M.S. Hacker, ‘Gordon Baker’s Late Interpretation of Wittgenstein’ in G. Kahane, E. Kanterian and O. Kuusela (eds.), Wittgenstein and His Interpreters: Essays in Memory of Gordon Baker (Oxford: Blackwell, 2007), p. 107.