Intellectual empathy

(English version)(Norwegian version here)

So now I suddenly had a blog, and people telling me to use it. What to do? Well I've always wanted to write down the pedagogical insight I have gained, to preserve it, especially now that I probably won't be giving lectures for the next few years. Let me today start with what I consider my most important discovery in pedagogy. Less complicated stuff will come in later posts.

So what is intellectual empathy? Well, other people have defined it, but that is not exactly what I understand by the phrase. With 'intellectual empathy' I understand the ability to imagine, or empathise with, other peoples intellectual grasp, or understanding, of the subject at hand. Let me try to give you some examples.

Foobar walks into your office and asks 'How does a computer really work?'. What will you tell him? (Assuming you know all about computers.) You need to know what he knows, whether he wants to make it 'shine some fancy coloured lights' or 'repartition the hard drive' or do some programming. If you start explaining about the computer without wanting to know what he already knows and what he wants to accomplish you are doomed to fail. Intellectual empathy is what drives you to find out what he knows, his background, and what he wants to accomplish with the information/understanding.

But this is just the first level of intellectual empathy. Most people are good at this, even though most people could spend a bit more time listening to the question before giving the answer. If the asker is after understanding and not just information, or fact, everything becomes more complicated.

Let me tell you about different levels of scientific understanding. On the most rudimentary level you know facts about the thing/science you are interested in. You know how to use a phone, the fact that it can communicate, with a delay of less than 1 second, with most people on the globe. On a more advanced level you know what the thing/science consist of. How to divide it in smaller parts and what these parts or ideas say. With understanding on an ever deeper level you know how to develop the thing/science itself, what the rules are for truth and experiment. This prepares us for the next example.

One of the things I remember well from my first semester at the university was in calculus 1 when I had no idea what the lecturer was talking about. In retrospect I see that they were proving the fundamental theorem of calculus, that differentiation is the opposite of integration (so that first integrating a function and then differentiating would give you back the function you started with). I remember they did lots of complicated stuff to prove something that I thought I already knew. What I already knew from high school was that integration is the opposite of differentiation and that it gives you the area under the graph. So, according to myself, I knew everything already.

My university lecturer understood the fundamental theorem well and had used it for years, he probably knew how it could be extended to more complicated ways of differentiating and integrating more advanced functions (as measures or distributions or stochastic integration for those who know these subjects), and found our use of this theorem to be quite trivial.

So where does intellectual empathy enter the picture? Being able to understand at which level of scientific understanding the subject is. What kind of understanding does a high school student have, how abstract is it? How procedural is it? How much is memorised and how much is actually understood at the intended level?

Let me tell you what levels the lecturer skipped in his presentations, what I needed to know before he could give me such an advanced lecture. First I needed to know what makes something true in mathematics, the idea of consistency. You can only have one definition, you can either define integration to be the opposite of differentiation, OR you can define integration to be the area under the graph. If you want to say that integration is ALSO the other thing you have to prove it.

The second thing I needed to know was how to read a proof. I was satisfied if the theorem (that which is to be proved) felt correct intuitively. But that, alas, is not sufficient for a proof, hence you can't just draw nice pictures or give examples illustrating the concept. A proof needs to be painstakingly obvious and logically flawless and precise.

The third thing I needed to know was why. Why should we care about this theorem? Is it just a fun exercise, or is it a fundamental property used in all of mathematics and physics?

So intellectual empathy is to look at the proof with your students eyes and try to understand what it says. If it doesn't make sense at all at their level of understanding you should talk about something else instead, as you are wasting everyone's time.

How can you train you intellectual empathy? I know of only two ways. The first is to ask the students to present the subject back you you just after you presented it to them, and if they do poorly try to figure out why. The other way is to help a below average student through the course material. And then I am NOT talking about two to three sessions per semester, but one to two long sessions every week with the same student. Then you will hopefully see their progression through the different levels of understanding; it can be an eye-opener.