Here’s a word problem for you to try:

Mary buys a stapler for $12, and also a notebook. She pays $17 altogether. Paul buys a notebook and also a ruler. He pays $8 less than Mary did. How much does the ruler cost?

Try to do it in your head, and pay attention to the mental gymnastics you go through as you solve it. In particular try to keep track of how you’re using your memory. Go ahead and give it a shot, I’ll wait.

Sips cognac, hums quietly to self; …checks watch

OK, If you’re like most people who attempt this problem your thought process went something like this:

  1. To find the cost of the ruler I need to figure out the cost of the notebook and the amount Paul paid.
  2. The cost of the notebook is the total Mary paid ($17) minus the cost of the stapler ($12), or $5.
  3. Paul paid $8 less than Mary ($17), or $9.
  4. The ruler is the amount Paul paid ($9) minus the cost of the notebook ($5), or $4.

The first thing you did was formulate a strategy. Then you systematically executed it. As you went through this process you had to perform several different groups of mental tasks, for example, determining the cost of the notebook. Cognitive scientists refer to these as task sets. Not only that, but you had to temporarily store the intermediate results of each task set in memory and then recall it for use in a subsequent step. You probably found the process relatively slow and effortful. In fact, you actually burned a few additional calories than normal while you were concentrating. If you were paying attention to your thought processes you could almost feel the intermediate quantities shuttling in and out of your working memory as you progressed.

If you’ve read my previous post on the dual process theory of cognition you probably recognized that you were mainly using your mental System 2 (i.e. reasoning) to solve the word problem above. Again, the defining characteristic of System 2 is that it is a serial process that uses working memory.As you parsed the text of the problem (largely a serial process) you formulated and stored your strategy. Then, referring to your strategy, you returned to the text to pick out the salient elements needed for each task set. You completed each task set in sequence, memorizing the intermediate results. For each subsequent task set you retrieved the intermediate result you needed and/or reparsed the text to get the next component of the problem. You continued in this way until you arrived at the solution.

One way you could have made it easier on yourself is to write the intermediate results down on paper. In fact, this was probably your immediate inclination. By storing the results in the environment, rather than in your head, you can save yourself considerable effort. The information you need to proceed is right there in front of you when you need it; you only need to shift your gaze to retrieve it. When you store it mentally you need to expend effort to put it there in the first place, to maintain it, and then to retrieve it. And the more items there are, the harder (and more error prone) the process is. Writing down the results of intermediate calculations saves you the effort of memorization. Note that it doesn’t save you the effort of performing the actual calculations, though.

Now, what if we represent the problem differently? In particular, what if we represent it graphically (that is, spatially), rather than linguistically? Below is a series of figures showing such an alternate representation. The quantities associated with the items involved are now represented as bars of varying lengths, organized by person. The bars are displayed relative to a metric ladder to make counting and length comparisons easier. Hash marks are used to indicate specific quantities. The given information is provided in fig. 1, that is, the people involved (Mary and Paul), the price Mary paid for the stapler ($12) and the total price she paid ($17).

Recall that in order to determine the cost of the ruler we first have to determine the cost of the notebook. This is easily and instantaneously read as the distance between the top of the bar representing the cost of the stapler and hash mark indicating the total amount paid by Mary (fig. 2). The second task in our strategy, determining the total price Paul paid, is just as easily found by counting down the given difference ($8) from the amount Mary paid (fig. 3). Finally, the price of the ruler is found by finding the difference in length between Paul’s total amount and the length of the bar representing the cost of the notebook (fig. 4). Note that these representations serve to remap the relatively abstract concept of differences in quantity to the concrete concept of differences in length.

Rerepresenting the problem in spatial form like this accelerates and simplifies problem solving in two key ways. First, as with the strategy of jotting things down on a piece of paper, the figures provide a form of artificial memory that can be used for storing and retrieving intermediate results. But beyond that, it also makes the computation process trivially easy. It does this by presenting the information in a way that almost entirely bypasses System 2 and directly engages System 1’s automatic and effortless spatial pattern processing engine. Why is this the case and how does it work? The answer will take some time to unspool, but it’s likely because we have evolved to think and reason largely in physical (e.g., spatial) terms. This is the embodied mind thesis, and, if correct, it offers a strong and principled path forward for advancing the field of human-information interface technology.