Enrico Persico, kids these days, and instrumental variables

Does sketching a problem really help solving it?

The full moon never rises at midnight. Can you tell whether this is true without making a sketch of the Sun-Earth-Moon system?

A while ago -and to my memory oddly synchronized with the terrible rotator-versus-wordcel meming spree that raged in some corners of the English language internet- I remember reading on Italian social media about an article by the physicist Enrico Persico. The title read –my translation- The importance of the physics that students dislike. The piece was a lamentation concerning the fact that students are not taking the time and effort to sketch a physical system before regurgitating a bunch of equations that supposedly describe it.

An excerpt, again in my translation: drawing, which would seem to be a spontaneous means of helping speech express what is in one's mind, is mostly not even considered by the student, and any invitation to make use of it is regarded as a cruelty. One gets the impression that the examinee lacks a mental image to put into words, but rather wants to merely repeat an argument as faithfully as possible.

Persico was part of the inner circle of Fermi, so the students we are talking about must be well into their eighties. Still, kids these days…

Anyway, Persico’s point of view is that this lack of sketching is a warning sign: has our future ruling class –comprised, back in his day, by students attending liceo, the high school that gave access to university education- lost interest in understanding physical phenomena in depth, feeling contented with the just-so story told by the blind application of mathematical formulae?

Galileo’s drawing of the phases of Venus

This seems a weird concern to have, but it is best put in perspective by remembering the tight relationship that sketching, drawing, and painting had with science during the Renaissance, when the ability to hand-draw realistic three-dimensional figures was a conditio sine qua non for recording and disseminating results in sciences ranging from anatomy to botany to astronomy. This lasted well into the modern age, with Galileo’s drawing of the phases of Venus dealing a devastating blow to geocentrism in 1610.

It took more than two centuries after that for astronomy to begin to rely on photographs, but even today, the simple act of drawing is still a powerful tool for understanding the geometry of objects we cannot see directly. For instance, the unified model of active galactic nuclei is one of the purest acts of geometrical imagination, very likely carried out by putting pencil to paper. Not to mention the importance of data visualization, which to this day is a key ingredient in many data analysis tasks and a close neighbor to interpretable machine learning in semantic space.

Back to Persico. What struck me in his argument is that he assumes the inability or unwillingness to sketch a physical situation to be a symptom of a fundamental gap in understanding. However it could be the other way round: the habit of drawing may lead to better understanding, so those who -for whatever reason- skip this step, end up having a harder time in solving physics problems. What is the direction of causation?

Luckily we do not need to speculate about this. We could easily settle the issue empirically with a sufficiently large group of students. What you need to do is to have a written examination with some sort of physics questions in it. Before the test starts, select half of your students at random and tell them to make a drawing of the system before writing down equations. Tell nothing to the other students who are -you guessed it- the control group. Or better yet give them some other unrelated hint. You could even have two versions of the assignment text. We will then have a TA evaluate the tests blindly, so we can check whether one group performed better than the other. Is this ethical? I do not know, but that is a question for another time.

Of course some students in the control group will sketch something anyway and some students that have been told to make a drawing won’t do it anyway. But this is not a problem: what we built by telling students to make a sketch is an instrumental variable –a variable that affects the outcome of the test only through the variable whose causal effect we want to estimate. In other words, this is a randomized controlled trial with an intention to treat framework. We know how to run those, so we can check whether Persico was right or not, provided we have enough statistical power.