27.1: Using Vision to Think

Pictures can Make Problems Easier

There is an ancient saying that a picture is worth a thousand words. Is it true? Well, it all depends on the picture, the words, and the problem we want to solve. But many problems that seem hopelessly confusing when described in words often become surprising clear when we find a way to represent them in a diagram. Indeed, with the right picture, a problem may pretty much solve itself; the answer just jumps out at us.

We have all heard about first cousins twice-removed, second-cousins once-removed, great uncles, and the like, but most of us aren’t very clear about what such relationships amount to; if we draw a family tree, however, things begin to fall into place. And the detailed workings of supply and demand and equilibrium in Econ 101 leave our heads spinning, but they begin to make sense when we learn to draw supply and demand graphs or curves. Indeed, there are many graphical or pictorial representations that make thinking easier: blueprints, seating charts, maps, flow diagrams and, increasingly, computer graphics and animations.

Figure \(\PageIndex{1}\): Bank Teller

Remember Linda, the outspoken, bright, philosophy major (16.4). We asked whether it is more probable (1) that she is a bank teller, or (2) that she is a bank teller who is active in the feminist movement. If we give the second answer, we commit the conjunction fallacy, but the problem can be confusing, and it may be difficult to see why this answer involves bad reasoning.

Once we draw a picture, though, the answer pops out at us. Here the crosshatched area where the circles overlap represents the set of bank tellers who are also feminists. Clearly this area cannot be larger than the entire circle on the left (which represents bank tellers in general) or than the entire circle on the right (which represents feminists in general).

Why are Pictures so Useful?

A large portion of the human brain (the visual cortex) is devoted to vision, and we are highly visual creatures. So, representing problems pictorially plays to our strengths.

One reason it is useful to present information pictorially is that this format helps compensate for limitations on working memory. We can’t focus directly on very much information at a time; our attention and working memory are very limited.

You can keep a six- or seven-digit number (like a phone number) in your working memory if you keep repeating it to yourself, but once you get to nine or ten digits it’s very difficult, and with twenty or so it’s hopeless.

Reasoning frequently requires us to focus on a good deal of information at once, and often we just can’t keep everything straight. Pictures can help. For one thing, they allow us to keep information that needs to be used together close together, so that we don’t have to keep searching around for the information we need.

Diagrams allow us to represent abstract relationships between individuals or sets with simple geometrical relationships like the inclusion of one circle in another (as in Figure 27.2.1) or the overlap of a pair of circles (as in the above diagram of feminist bank tellers). We can also represent facts about percentages and proportions (which is often a good way to think about probabilities) by the relative size of different parts of a diagram, as we do in a bar graph or pie chart (see Figure 27.3.3).

Such representations are useful because humans are very good at recognizing geometrical relationships like overlap, inclusion, and relative size. When we exploit such structure in a diagram, the information is organized in a way that we can take in at a glance, and often we can draw visual inferences almost automatically.

Diagrams are also useful for exploration and communication. When we draw a diagram, we can tinker with it, erasing the circle here, adding a dot over there, patching it up by trial and error as we grope our way to clarity. And in many cases, it is easier to communicate an idea with a diagram than with words.

Different situations and problems call for different sorts of diagrams, and some problems aren’t usefully represented by diagrams at all. Furthermore, diagrams, like all representations, can distort, muddy, and confuse. Still, when we can draw a good diagram, it often cuts a very confusing problem down to a manageable size.

In this chapter, we will get a feel for the sorts of problems that can be tackled with diagrams, and learn something about which sorts of diagrams are useful where. The goal is to add diagramming to the set of cognitive tools that you can use, so we will focus on simple diagrams, ones you can often construct in a matter of seconds on the back of a napkin.